Theorem infxpidmlem8 4940

Description: Lemma for infxpidm 4945. The union of a collection of chains C in the collection of bijections H belongs to H. This property will be needed to apply Zorn's Lemma in infxpidmlem9 4941.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem6.2	|- B = ran U.C
infxpidmlem8.3	|- C e. V

Assertion

Ref	Expression
infxpidmlem8	|- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

Distinct variable group(s):   f,g,h,t,A   B,f,g,h,t   C,f,g,h,t   g,H,h

Proof of Theorem infxpidmlem8

Step	Hyp	Ref	Expression
1		ssel2 1503	. . . . . . . . . . 11 |- ((C (_ H /\ g e. C) -> g e. H)
2		infxpidmlem.1	. . . . . . . . . . . . . . 15 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3		visset 1350	. . . . . . . . . . . . . . 15 |- g e. V
4	2, 3	infxpidmlem2 4934	. . . . . . . . . . . . . 14 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
5	4	biimp 133	. . . . . . . . . . . . 13 |- (g e. H -> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
6	5	ord 202	. . . . . . . . . . . 12 |- (g e. H -> (-. g = (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
7		f1ofo 2806	. . . . . . . . . . . . . . . . . 18 |- (g:(x X. x)-1-1-onto->x -> g:(x X. x)-onto->x)
8		forn 2789	. . . . . . . . . . . . . . . . . 18 |- (g:(x X. x)-onto->x -> ran g = x)
9	7, 8	syl 12	. . . . . . . . . . . . . . . . 17 |- (g:(x X. x)-1-1-onto->x -> ran g = x)
10	9	cleqcomd 1106	. . . . . . . . . . . . . . . 16 |- (g:(x X. x)-1-1-onto->x -> x = ran g)
11	10	anim1i 269	. . . . . . . . . . . . . . 15 |- ((g:(x X. x)-1-1-onto->x /\ (om ~<_ x /\ x (_ A)) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
12	11	ancoms 334	. . . . . . . . . . . . . 14 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (x = ran g /\ (om ~<_ x /\ x (_ A)))
13	12	19.22i 723	. . . . . . . . . . . . 13 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> E.x(x = ran g /\ (om ~<_ x /\ x (_ A)))
14		rnexg 2569	. . . . . . . . . . . . . . 15 |- (g e. V -> ran g e. V)
15	3, 14	ax-mp 6	. . . . . . . . . . . . . 14 |- ran g e. V
16		breq2 2066	. . . . . . . . . . . . . . 15 |- (x = ran g -> (om ~<_ x <-> om ~<_ ran g))
17		sseq1 1521	. . . . . . . . . . . . . . 15 |- (x = ran g -> (x (_ A <-> ran g (_ A))
18	16, 17	anbi12d 476	. . . . . . . . . . . . . 14 |- (x = ran g -> ((om ~<_ x /\ x (_ A) <-> (om ~<_ ran g /\ ran g (_ A)))
19	15, 18	ceqsexv 1371	. . . . . . . . . . . . 13 |- (E.x(x = ran g /\ (om ~<_ x /\ x (_ A)) <-> (om ~<_ ran g /\ ran g (_ A))
20	13, 19	sylib 173	. . . . . . . . . . . 12 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ ran g /\ ran g (_ A))
21	6, 20	syl6 23	. . . . . . . . . . 11 |- (g e. H -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
22	1, 21	syl 12	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> (om ~<_ ran g /\ ran g (_ A)))
23		domtr 3320	. . . . . . . . . . . . . . 15 |- ((om ~<_ ran g /\ ran g ~<_ B) -> om ~<_ B)
24		ra4e 1244	. . . . . . . . . . . . . . . . . . 19 |- ((g e. C /\ y e. ran g) -> E.g e. C y e. ran g)
25		infxpidmlem6.2	. . . . . . . . . . . . . . . . . . . 20 |- B = ran U.C
26	2, 25	infxpidmlem6 4938	. . . . . . . . . . . . . . . . . . 19 |- (y e. B <-> E.g e. C y e. ran g)
27	24, 26	sylibr 175	. . . . . . . . . . . . . . . . . 18 |- ((g e. C /\ y e. ran g) -> y e. B)
28	27	exp 291	. . . . . . . . . . . . . . . . 17 |- (g e. C -> (y e. ran g -> y e. B))
29	28	ssrdv 1509	. . . . . . . . . . . . . . . 16 |- (g e. C -> ran g (_ B)
30		ssdomg 3311	. . . . . . . . . . . . . . . . 17 |- (ran g e. V -> (ran g (_ B -> ran g ~<_ B))
31	15, 30	ax-mp 6	. . . . . . . . . . . . . . . 16 |- (ran g (_ B -> ran g ~<_ B)
32	29, 31	syl 12	. . . . . . . . . . . . . . 15 |- (g e. C -> ran g ~<_ B)
33	23, 32	sylan2 346	. . . . . . . . . . . . . 14 |- ((om ~<_ ran g /\ g e. C) -> om ~<_ B)
34	33	exp 291	. . . . . . . . . . . . 13 |- (om ~<_ ran g -> (g e. C -> om ~<_ B))
35	34	com12 13	. . . . . . . . . . . 12 |- (g e. C -> (om ~<_ ran g -> om ~<_ B))
36	35	adantl 305	. . . . . . . . . . 11 |- ((C (_ H /\ g e. C) -> (om ~<_ ran g -> om ~<_ B))
37	36	adantrd 308	. . . . . . . . . 10 |- ((C (_ H /\ g e. C) -> ((om ~<_ ran g /\ ran g (_ A) -> om ~<_ B))
38	22, 37	syld 27	. . . . . . . . 9 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> om ~<_ B))
39	38	exp 291	. . . . . . . 8 |- (C (_ H -> (g e. C -> (-. g = (/) -> om ~<_ B)))
40	39	r19.23adv 1286	. . . . . . 7 |- (C (_ H -> (E.g e. C -. g = (/) -> om ~<_ B))
41		uni0b 1939	. . . . . . . . . 10 |- (U.C = (/) <-> C (_ {(/)})
42		dfss3 1498	. . . . . . . . . 10 |- (C (_ {(/)} <-> A.g e. C g e. {(/)})
43		elsn 1820	. . . . . . . . . . 11 |- (g e. {(/)} <-> g = (/))
44	43	biral 1223	. . . . . . . . . 10 |- (A.g e. C g e. {(/)} <-> A.g e. C g = (/))
45	41, 42, 44	3bitr 155	. . . . . . . . 9 |- (U.C = (/) <-> A.g e. C g = (/))
46	45	negbii 162	. . . . . . . 8 |- (-. U.C = (/) <-> -. A.g e. C g = (/))
47		rexnal 1210	. . . . . . . 8 |- (E.g e. C -. g = (/) <-> -. A.g e. C g = (/))
48	46, 47	bitr4 154	. . . . . . 7 |- (-. U.C = (/) <-> E.g e. C -. g = (/))
49	40, 48	syl5ib 181	. . . . . 6 |- (C (_ H -> (-. U.C = (/) -> om ~<_ B))
50		pm3.27 260	. . . . . . . . . . . . . 14 |- ((om ~<_ ran g /\ ran g (_ A) -> ran g (_ A)
51	22, 50	syl6 23	. . . . . . . . . . . . 13 |- ((C (_ H /\ g e. C) -> (-. g = (/) -> ran g (_ A))
52		rneq 2555	. . . . . . . . . . . . . . 15 |- (g = (/) -> ran g = ran (/))
53		rn0 2567	. . . . . . . . . . . . . . 15 |- ran (/) = (/)
54	52, 53	syl6eq 1140	. . . . . . . . . . . . . 14 |- (g = (/) -> ran g = (/))
55		0ss 1725	. . . . . . . . . . . . . . 15 |- (/) (_ A
56	55	a1i 7	. . . . . . . . . . . . . 14 |- (g = (/) -> (/) (_ A)
57	54, 56	eqsstrd 1534	. . . . . . . . . . . . 13 |- (g = (/) -> ran g (_ A)
58	51, 57	pm2.61d2 111	. . . . . . . . . . . 12 |- ((C (_ H /\ g e. C) -> ran g (_ A)
59	58	sseld 1506	. . . . . . . . . . 11 |- ((C (_ H /\ g e. C) -> (y e. ran g -> y e. A))
60	59	exp 291	. . . . . . . . . 10 |- (C (_ H -> (g e. C -> (y e. ran g -> y e. A)))
61	60	r19.23adv 1286	. . . . . . . . 9 |- (C (_ H -> (E.g e. C y e. ran g -> y e. A))
62	61, 26	syl5ib 181	. . . . . . . 8 |- (C (_ H -> (y e. B -> y e. A))
63	62	ssrdv 1509	. . . . . . 7 |- (C (_ H -> B (_ A)
64	63	a1d 14	. . . . . 6 |- (C (_ H -> (-. U.C = (/) -> B (_ A))
65	49, 64	jcad 455	. . . . 5 |- (C (_ H -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
66	65	adantr 306	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> (om ~<_ B /\ B (_ A)))
67	2, 25	infxpidmlem7 4939	. . . . 5 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C:(B X. B)-1-1-onto->B)
68	67	a1d 14	. . . 4 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> U.C:(B X. B)-1-1-onto->B))
69	66, 68	jcad 455	. . 3 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> ((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B)))
70		infxpidmlem8.3	. . . . 5 |- C e. V
71	70	uniex 1947	. . . 4 |- U.C e. V
72		rnexg 2569	. . . . . 6 |- (U.C e. V -> ran U.C e. V)
73	71, 72	ax-mp 6	. . . . 5 |- ran U.C e. V
74	25, 73	eqeltr 1159	. . . 4 |- B e. V
75	2, 71, 74	infxpidmlem3 4935	. . 3 |- (((om ~<_ B /\ B (_ A) /\ U.C:(B X. B)-1-1-onto->B) -> U.C e. H)
76	69, 75	syl6 23	. 2 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> (-. U.C = (/) -> U.C e. H))
77		orc 225	. . 3 |- (U.C = (/) -> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
78	2, 71	infxpidmlem2 4934	. . 3 |- (U.C e. H <-> (U.C = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ U.C:(x X. x)-1-1-onto->x)))
79	77, 78	sylibr 175	. 2 |- (U.C = (/) -> U.C e. H)
80	76, 79	pm2.61d2 111	1 |- ((C (_ H /\ A.g e. C A.h e. C (g (_ h \/ h (_ g)) -> U.C e. H)

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  (/)c0 1707  {csn 1808  U.cuni 1919   class class class wbr 2054  omcom 2372   X. cxp 2408  ran crn 2411  -onto->wfo 2420  -1-1-onto->wf1o 2421   ~<_ cdom 3272

This theorem is referenced by:  infxpidmlem9 4941

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428