Theorem infxpidmlem4 4936

Description: Lemma for infxpidm 4945. The domain of a member of H is the cross product of its range.

Hypothesis

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

Assertion

Ref	Expression
infxpidmlem4	|- (g e. H -> dom g = (ran g X. ran g))

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

Proof of Theorem infxpidmlem4

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . 3 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
2		visset 1350	. . 3 |- g e. V
3	1, 2	infxpidmlem2 4934	. 2 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
4		dmeq 2531	. . . . 5 |- (g = (/) -> dom g = dom (/))
5		dm0 2542	. . . . 5 |- dom (/) = (/)
6	4, 5	syl6eq 1140	. . . 4 |- (g = (/) -> dom g = (/))
7		rneq 2555	. . . . . . 7 |- (g = (/) -> ran g = ran (/))
8		rn0 2567	. . . . . . 7 |- ran (/) = (/)
9	7, 8	syl6eq 1140	. . . . . 6 |- (g = (/) -> ran g = (/))
10		xpeq2 2441	. . . . . 6 |- (ran g = (/) -> (ran g X. ran g) = (ran g X. (/)))
11	9, 10	syl 12	. . . . 5 |- (g = (/) -> (ran g X. ran g) = (ran g X. (/)))
12		xp0 2652	. . . . 5 |- (ran g X. (/)) = (/)
13	11, 12	syl6eq 1140	. . . 4 |- (g = (/) -> (ran g X. ran g) = (/))
14	6, 13	eqtr4d 1131	. . 3 |- (g = (/) -> dom g = (ran g X. ran g))
15		f1o2 2804	. . . . . 6 |- (g:(x X. x)-1-1-onto->x <-> (g Fn (x X. x) /\ Fun `'g /\ ran g = x))
16		fndm 2723	. . . . . . . 8 |- (g Fn (x X. x) -> dom g = (x X. x))
17		xpeq1 2440	. . . . . . . . 9 |- (ran g = x -> (ran g X. ran g) = (x X. ran g))
18		xpeq2 2441	. . . . . . . . 9 |- (ran g = x -> (x X. ran g) = (x X. x))
19	17, 18	eqtr2d 1129	. . . . . . . 8 |- (ran g = x -> (x X. x) = (ran g X. ran g))
20	16, 19	sylan9eq 1144	. . . . . . 7 |- ((g Fn (x X. x) /\ ran g = x) -> dom g = (ran g X. ran g))
21	20	3adant2 598	. . . . . 6 |- ((g Fn (x X. x) /\ Fun `'g /\ ran g = x) -> dom g = (ran g X. ran g))
22	15, 21	sylbi 174	. . . . 5 |- (g:(x X. x)-1-1-onto->x -> dom g = (ran g X. ran g))
23	22	adantl 305	. . . 4 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> dom g = (ran g X. ran g))
24	23	19.23aiv 952	. . 3 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> dom g = (ran g X. ran g))
25	14, 24	jaoi 275	. 2 |- ((g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)) -> dom g = (ran g X. ran g))
26	3, 25	sylbi 174	1 |- (g e. H -> dom g = (ran g X. ran g))

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196   /\ w3a 581  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707   class class class wbr 2054  omcom 2372   X. cxp 2408  `'ccnv 2409  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  -1-1-onto->wf1o 2421   ~<_ cdom 3272

This theorem is referenced by:  infxpidmlem5 4937  infxpidmlem7 4939

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-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-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-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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