Theorem infxpidmlem9 4941

Description: Lemma for infxpidm 4945. By Zorn's Lemma zorn2 3612, the collection H (which we show here to be a set) has a maximal element.

Hypotheses

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

Assertion

Ref	Expression
infxpidmlem9	|- E.g e. H A.h e. H -. g (. h

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

Proof of Theorem infxpidmlem9

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . . . 5 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
2		unab 1691	. . . . 5 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3	1, 2	eqtr4 1122	. . . 4 |- H = ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)})
4		df-sn 1811	. . . . . 6 |- {(/)} = {f | f = (/)}
5		p0ex 1885	. . . . . 6 |- {(/)} e. V
6	4, 5	eqeltrr 1160	. . . . 5 |- {f | f = (/)} e. V
7		df-rex 1206	. . . . . . . 8 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
8		visset 1350	. . . . . . . . . . . 12 |- t e. V
9	8	elpw 1801	. . . . . . . . . . 11 |- (t e. P~A <-> t (_ A)
10	9	anbi1i 368	. . . . . . . . . 10 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> (t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)))
11		ancom 333	. . . . . . . . . 10 |- ((t (_ A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A))
12		an23 371	. . . . . . . . . 10 |- (((om ~<_ t /\ f:(t X. t)-1-1-onto->t) /\ t (_ A) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
13	10, 11, 12	3bitr 155	. . . . . . . . 9 |- ((t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> ((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
14	13	biex 733	. . . . . . . 8 |- (E.t(t e. P~A /\ (om ~<_ t /\ f:(t X. t)-1-1-onto->t)) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
15	7, 14	bitr 151	. . . . . . 7 |- (E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t) <-> E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))
16	15	biabi 1181	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} = {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}
17		infxpidmlem.2	. . . . . . . 8 |- A e. V
18	17	pwex 1806	. . . . . . 7 |- P~A e. V
19	8, 8	xpex 2488	. . . . . . . . 9 |- (t X. t) e. V
20		mapex 3261	. . . . . . . . 9 |- (((t X. t) e. V /\ t e. V) -> {f | f:(t X. t)-->t} e. V)
21	19, 8, 20	mp2an 520	. . . . . . . 8 |- {f | f:(t X. t)-->t} e. V
22		f1of 2800	. . . . . . . . . 10 |- (f:(t X. t)-1-1-onto->t -> f:(t X. t)-->t)
23	22	adantl 305	. . . . . . . . 9 |- ((om ~<_ t /\ f:(t X. t)-1-1-onto->t) -> f:(t X. t)-->t)
24	23	ss2abi 1552	. . . . . . . 8 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} (_ {f | f:(t X. t)-->t}
25	21, 24	ssexi 1701	. . . . . . 7 |- {f | (om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
26	18, 25	abrexex2 2915	. . . . . 6 |- {f | E.t e. P~ A(om ~<_ t /\ f:(t X. t)-1-1-onto->t)} e. V
27	16, 26	eqeltrr 1160	. . . . 5 |- {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)} e. V
28	6, 27	unex 1949	. . . 4 |- ({f | f = (/)} u. {f | E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t)}) e. V
29	3, 28	eqeltr 1159	. . 3 |- H e. V
30	29	zorn2 3612	. 2 |- (A.z((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H) -> E.g e. H A.h e. H -. g (. h)
31		cleqid 1102	. . 3 |- ran U.z = ran U.z
32		visset 1350	. . 3 |- z e. V
33	1, 31, 32	infxpidmlem8 4940	. 2 |- ((z (_ H /\ A.g e. z A.h e. z (g (_ h \/ h (_ g)) -> U.z e. H)
34	30, 33	mpg 684	1 |- E.g e. H A.h e. H -. g (. 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   u. cun 1485   (_ wss 1487   (. wpss 1488  (/)c0 1707  P~cpw 1798  {csn 1808  U.cuni 1919   class class class wbr 2054  omcom 2372   X. cxp 2408  ran crn 2411  -->wf 2418  -1-1-onto->wf1o 2421   ~<_ cdom 3272

This theorem is referenced by:  infxpidmlem12 4944

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  ax-reg 1078  ax-ac 1080

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-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  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-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439  df-en 3274  df-dom 3275

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