Theorem infxpidmlem5 4937

Description: Lemma for infxpidm 4945. Two members in the range of a member of a subset of H form an ordered pair belonging to the domain of the union of the subset.

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
infxpidmlem5	|- ((C (_ H /\ g e. C) -> ((y e. ran g /\ z e. ran g) -> <.y, z>. e. dom U.C))

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

Proof of Theorem infxpidmlem5

Step	Hyp	Ref	Expression
1		ssel2 1503	. . . . 5 |- ((C (_ H /\ g e. C) -> g e. H)
2		infxpidmlem.1	. . . . . 6 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
3	2	infxpidmlem4 4936	. . . . 5 |- (g e. H -> dom g = (ran g X. ran g))
4	1, 3	syl 12	. . . 4 |- ((C (_ H /\ g e. C) -> dom g = (ran g X. ran g))
5	4	eleq2d 1156	. . 3 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> <.y, z>. e. (ran g X. ran g)))
6		visset 1350	. . . 4 |- z e. V
7	6	opelxp 2452	. . 3 |- (<.y, z>. e. (ran g X. ran g) <-> (y e. ran g /\ z e. ran g))
8	5, 7	syl6bb 414	. 2 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> (y e. ran g /\ z e. ran g)))
9		ssiun2 2019	. . . . 5 |- (g e. C -> dom g (_ U.g e. C dom g)
10		dmuni 2538	. . . . 5 |- dom U.C = U.g e. C dom g
11	9, 10	syl6ssr 1547	. . . 4 |- (g e. C -> dom g (_ dom U.C)
12	11	sseld 1506	. . 3 |- (g e. C -> (<.y, z>. e. dom g -> <.y, z>. e. dom U.C))
13	12	adantl 305	. 2 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g -> <.y, z>. e. dom U.C))
14	8, 13	sylbird 180	1 |- ((C (_ H /\ g e. C) -> ((y e. ran g /\ z e. ran g) -> <.y, z>. e. dom U.C))

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  <.cop 1810  U.cuni 1919  U.ciun 1994   class class class wbr 2054  omcom 2372   X. cxp 2408  dom cdm 2410  ran crn 2411  -1-1-onto->wf1o 2421   ~<_ cdom 3272

This theorem is referenced by:  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-rex 1206  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-uni 1920  df-iun 1996  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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