Theorem infxpidmlem4 7555

Description: Lemma for infxpidm 7564. 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 1813	. . 3 |- g e. V
3	1, 2	infxpidmlem2 7553	. 2 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
4		dm0 3323	. . . 4 |- dom (/) = (/)
5		dmeq 3311	. . . 4 |- (g = (/) -> dom g = dom (/))
6		rneq 3339	. . . . . . 7 |- (g = (/) -> ran g = ran (/))
7		rn0 3355	. . . . . . 7 |- ran (/) = (/)
8	6, 7	syl6eq 1523	. . . . . 6 |- (g = (/) -> ran g = (/))
9		xpeq2 3201	. . . . . 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 3465	. . . . 5 |- (ran g X. (/)) = (/)
12	10, 11	syl6eq 1523	. . . 4 |- (g = (/) -> (ran g X. ran g) = (/))
13	4, 5, 12	3eqtr4a 1532	. . 3 |- (g = (/) -> dom g = (ran g X. ran g))
14		f1o2 3693	. . . . . 6 |- (g:(x X. x)-1-1-onto->x <-> (g Fn (x X. x) /\ Fun `'g /\ ran g = x))
15		fndm 3587	. . . . . . . 8 |- (g Fn (x X. x) -> dom g = (x X. x))
16		xpeq1 3200	. . . . . . . . 9 |- (ran g = x -> (ran g X. ran g) = (x X. ran g))
17		xpeq2 3201	. . . . . . . . 9 |- (ran g = x -> (x X. ran g) = (x X. x))
18	16, 17	eqtr2d 1508	. . . . . . . 8 |- (ran g = x -> (x X. x) = (ran g X. ran g))
19	15, 18	sylan9eq 1527	. . . . . . 7 |- ((g Fn (x X. x) /\ ran g = x) -> dom g = (ran g X. ran g))
20	19	3adant2 798	. . . . . 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 388	. . . 4 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> dom g = (ran g X. ran g))
23	22	19.23aiv 1295	. . 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 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   (_ wss 2047  (/)c0 2280   class class class wbr 2619  omcom 3131   X. cxp 3168  `'ccnv 3169  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -1-1-onto->wf1o 3181   ~<_ cdom 4365

This theorem is referenced by:  infxpidmlem5 7556  infxpidmlem7 7558

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

Copyright terms: Public domain