Theorem infxpidmlem4 7556

Description: Lemma for infxpidm 7565. 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 groups:   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 1816	. . 3 |- g e. V
3	1, 2	infxpidmlem2 7554	. 2 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
4		dm0 3329	. . . 4 |- dom (/) = (/)
5		dmeq 3317	. . . 4 |- (g = (/) -> dom g = dom (/))
6		rneq 3345	. . . . . . 7 |- (g = (/) -> ran g = ran (/))
7		rn0 3361	. . . . . . 7 |- ran (/) = (/)
8	6, 7	syl6eq 1526	. . . . . 6 |- (g = (/) -> ran g = (/))
9		xpeq2 3207	. . . . . 6 |- (ran g = (/) -> (ran g X. ran g) = (ran g X. (/)))
10	8, 9	syl 10	. . . . 5 |- (g = (/) -> (ran g X. ran g) = (ran g X. (/)))
11		xp0 3471	. . . . 5 |- (ran g X. (/)) = (/)
12	10, 11	syl6eq 1526	. . . 4 |- (g = (/) -> (ran g X. ran g) = (/))
13	4, 5, 12	3eqtr4a 1535	. . 3 |- (g = (/) -> dom g = (ran g X. ran g))
14		f1o2 3699	. . . . . 6 |- (g:(x X. x)-1-1-onto->x <-> (g Fn (x X. x) /\ Fun `'g /\ ran g = x))
15		fndm 3593	. . . . . . . 8 |- (g Fn (x X. x) -> dom g = (x X. x))
16		xpeq1 3206	. . . . . . . . 9 |- (ran g = x -> (ran g X. ran g) = (x X. ran g))
17		xpeq2 3207	. . . . . . . . 9 |- (ran g = x -> (x X. ran g) = (x X. x))
18	16, 17	eqtr2d 1511	. . . . . . . 8 |- (ran g = x -> (x X. x) = (ran g X. ran g))
19	15, 18	sylan9eq 1530	. . . . . . 7 |- ((g Fn (x X. x) /\ ran g = x) -> dom g = (ran g X. ran g))
20	19	3adant2 800	. . . . . 6 |- ((g Fn (x X. x) /\ Fun `'g /\ ran g = x) -> dom g = (ran g X. ran g))
21	14, 20	sylbi 199	. . . . 5 |- (g:(x X. x)-1-1-onto->x -> dom g = (ran g X. ran g))
22	21	adantl 390	. . . 4 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> dom g = (ran g X. ran g))
23	22	19.23aiv 1297	. . 3 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> dom g = (ran g X. ran g))
24	13, 23	jaoi 341	. 2 |- ((g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)) -> dom g = (ran g X. ran g))
25	3, 24	sylbi 199	1 |- (g e. H -> dom g = (ran g X. ran g))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   (_ wss 2050  (/)c0 2283   class class class wbr 2624  omcom 3137   X. cxp 3174  `'ccnv 3175  dom cdm 3176  ran crn 3177  Fun wfun 3182   Fn wfn 3183  -1-1-onto->wf1o 3187   ~<_ cdom 4371

This theorem is referenced by:  infxpidmlem5 7557  infxpidmlem7 7559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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