Theorem tfrlem10 2958

Description: Lemma for transfinite recursion. We define class C by extending F with one ordered pair. We will assume, falsely, that domain of F is a member of, and thus not equal to, On. Using this assumption we will prove facts about C that will lead to a contradiction in tfrlem13 2961, thus showing the domain of F does in fact equal On. Here we show (under the false assumption) that C is a function extending the domain of F by one.

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A
tfrlem.3	|- C = (F u. {<.dom F, (G` (F |` dom F))>.})

Assertion

Ref	Expression
tfrlem10	|- (dom F e. On -> C Fn suc dom F)

Distinct variable group(s):   x,y,f,A   x,F,y,f   x,G,y,f   x,C,y,f

Proof of Theorem tfrlem10

Step	Hyp	Ref	Expression
1		opeq1 1876	. . . . . 6 |- (x = dom F -> <.x, (G` (F |` dom F))>. = <.dom F, (G` (F |` dom F))>.)
2		sneq 1816	. . . . . 6 |- (<.x, (G` (F |` dom F))>. = <.dom F, (G` (F |` dom F))>. -> {<.x, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.})
3		funeq 2683	. . . . . 6 |- ({<.x, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.} -> (Fun {<.x, (G` (F |` dom F))>.} <-> Fun {<.dom F, (G` (F |` dom F))>.}))
4	1, 2, 3	3syl 21	. . . . 5 |- (x = dom F -> (Fun {<.x, (G` (F |` dom F))>.} <-> Fun {<.dom F, (G` (F |` dom F))>.}))
5		visset 1350	. . . . . 6 |- x e. V
6		fvex 2838	. . . . . 6 |- (G` (F |` dom F)) e. V
7	5, 6	funsn 2690	. . . . 5 |- Fun {<.x, (G` (F |` dom F))>.}
8	4, 7	vtoclg 1383	. . . 4 |- (dom F e. On -> Fun {<.dom F, (G` (F |` dom F))>.})
9		tfrlem.1	. . . . . 6 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
10		tfrlem.2	. . . . . 6 |- F = U.A
11	9, 10	tfrlem7 2955	. . . . 5 |- Fun F
12		funun 2700	. . . . . . . 8 |- (((Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}) /\ (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/)) -> Fun (F u. {<.dom F, (G` (F |` dom F))>.}))
13		tfrlem.3	. . . . . . . . 9 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
14		funeq 2683	. . . . . . . . 9 |- (C = (F u. {<.dom F, (G` (F |` dom F))>.}) -> (Fun C <-> Fun (F u. {<.dom F, (G` (F |` dom F))>.})))
15	13, 14	ax-mp 6	. . . . . . . 8 |- (Fun C <-> Fun (F u. {<.dom F, (G` (F |` dom F))>.}))
16	12, 15	sylibr 175	. . . . . . 7 |- (((Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}) /\ (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/)) -> Fun C)
17	1	sneqd 1818	. . . . . . . . . . . . . . 15 |- (x = dom F -> {<.x, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.})
18	17	dmeqd 2533	. . . . . . . . . . . . . 14 |- (x = dom F -> dom {<.x, (G` (F |` dom F))>.} = dom {<.dom F, (G` (F |` dom F))>.})
19		sneq 1816	. . . . . . . . . . . . . 14 |- (x = dom F -> {x} = {dom F})
20	18, 19	cleq12d 1115	. . . . . . . . . . . . 13 |- (x = dom F -> (dom {<.x, (G` (F |` dom F))>.} = {x} <-> dom {<.dom F, (G` (F |` dom F))>.} = {dom F}))
21		dmsnop 2547	. . . . . . . . . . . . 13 |- dom {<.x, (G` (F |` dom F))>.} = {x}
22	20, 21	vtoclg 1383	. . . . . . . . . . . 12 |- (dom F e. On -> dom {<.dom F, (G` (F |` dom F))>.} = {dom F})
23	22	eleq2d 1156	. . . . . . . . . . 11 |- (dom F e. On -> (x e. dom {<.dom F, (G` (F |` dom F))>.} <-> x e. {dom F}))
24		eleq1 1149	. . . . . . . . . . . . . . 15 |- (x = dom F -> (x e. dom F <-> dom F e. dom F))
25	24	negbid 463	. . . . . . . . . . . . . 14 |- (x = dom F -> (-. x e. dom F <-> -. dom F e. dom F))
26		eloni 2209	. . . . . . . . . . . . . . 15 |- (dom F e. On -> Ord dom F)
27		ordeirr 2217	. . . . . . . . . . . . . . 15 |- (Ord dom F -> -. dom F e. dom F)
28	26, 27	syl 12	. . . . . . . . . . . . . 14 |- (dom F e. On -> -. dom F e. dom F)
29	25, 28	syl5bir 184	. . . . . . . . . . . . 13 |- (x = dom F -> (dom F e. On -> -. x e. dom F))
30	29	com12 13	. . . . . . . . . . . 12 |- (dom F e. On -> (x = dom F -> -. x e. dom F))
31		elsni 1827	. . . . . . . . . . . 12 |- (x e. {dom F} -> x = dom F)
32	30, 31	syl5 22	. . . . . . . . . . 11 |- (dom F e. On -> (x e. {dom F} -> -. x e. dom F))
33	23, 32	sylbid 178	. . . . . . . . . 10 |- (dom F e. On -> (x e. dom {<.dom F, (G` (F |` dom F))>.} -> -. x e. dom F))
34	33	con2d 83	. . . . . . . . 9 |- (dom F e. On -> (x e. dom F -> -. x e. dom {<.dom F, (G` (F |` dom F))>.}))
35	34	r19.21aiv 1259	. . . . . . . 8 |- (dom F e. On -> A.x e. dom F -. x e. dom {<.dom F, (G` (F |` dom F))>.})
36		disj 1733	. . . . . . . 8 |- ((dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/) <-> A.x e. dom F -. x e. dom {<.dom F, (G` (F |` dom F))>.})
37	35, 36	sylibr 175	. . . . . . 7 |- (dom F e. On -> (dom F i^i dom {<.dom F, (G` (F |` dom F))>.}) = (/))
38	16, 37	sylan2 346	. . . . . 6 |- (((Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}) /\ dom F e. On) -> Fun C)
39	38	exp 291	. . . . 5 |- ((Fun F /\ Fun {<.dom F, (G` (F |` dom F))>.}) -> (dom F e. On -> Fun C))
40	11, 39	mpan 518	. . . 4 |- (Fun {<.dom F, (G` (F |` dom F))>.} -> (dom F e. On -> Fun C))
41	8, 40	mpcom 49	. . 3 |- (dom F e. On -> Fun C)
42		dmeq 2531	. . . . . . . 8 |- ({<.x, (G` (F |` dom F))>.} = {<.dom F, (G` (F |` dom F))>.} -> dom {<.x, (G` (F |` dom F))>.} = dom {<.dom F, (G` (F |` dom F))>.})
43	1, 2, 42	3syl 21	. . . . . . 7 |- (x = dom F -> dom {<.x, (G` (F |` dom F))>.} = dom {<.dom F, (G` (F |` dom F))>.})
44	43, 19	cleq12d 1115	. . . . . 6 |- (x = dom F -> (dom {<.x, (G` (F |` dom F))>.} = {x} <-> dom {<.dom F, (G` (F |` dom F))>.} = {dom F}))
45	44, 21	vtoclg 1383	. . . . 5 |- (dom F e. On -> dom {<.dom F, (G` (F |` dom F))>.} = {dom F})
46	45	uneq2d 1611	. . . 4 |- (dom F e. On -> (dom F u. dom {<.dom F, (G` (F |` dom F))>.}) = (dom F u. {dom F}))
47	13	dmeqi 2532	. . . . 5 |- dom C = dom (F u. {<.dom F, (G` (F |` dom F))>.})
48		dmun 2536	. . . . 5 |- dom (F u. {<.dom F, (G` (F |` dom F))>.}) = (dom F u. dom {<.dom F, (G` (F |` dom F))>.})
49	47, 48	eqtr 1119	. . . 4 |- dom C = (dom F u. dom {<.dom F, (G` (F |` dom F))>.})
50		df-suc 2205	. . . 4 |- suc dom F = (dom F u. {dom F})
51	46, 49, 50	3eqtr4g 1147	. . 3 |- (dom F e. On -> dom C = suc dom F)
52	41, 51	jca 236	. 2 |- (dom F e. On -> (Fun C /\ dom C = suc dom F))
53		df-fn 2433	. 2 |- (C Fn suc dom F <-> (Fun C /\ dom C = suc dom F))
54	52, 53	sylibr 175	1 |- (dom F e. On -> C Fn suc dom F)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   u. cun 1485   i^i cin 1486  (/)c0 1707  {csn 1808  <.cop 1810  U.cuni 1919  Ord word 2198  Oncon0 2199  suc csuc 2201  dom cdm 2410   |` cres 2412  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  tfrlem11 2959  tfrlem12 2960  tfrlem13 2961

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-3or 582  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-rab 1208  df-v 1349  df-sbc 1441  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-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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