Theorem infxpidmlem5 7516

Description: Lemma for infxpidm 7524. 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 groups:   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 2061	. . . . 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 7515	. . . . 5 |- (g e. H -> dom g = (ran g X. ran g))
4	1, 3	syl 10	. . . 4 |- ((C (_ H /\ g e. C) -> dom g = (ran g X. ran g))
5	4	eleq2d 1539	. . 3 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> <.y, z>. e. (ran g X. ran g)))
6		visset 1810	. . . 4 |- z e. V
7	6	opelxp 3210	. . 3 |- (<.y, z>. e. (ran g X. ran g) <-> (y e. ran g /\ z e. ran g))
8	5, 7	syl6bb 535	. 2 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> (y e. ran g /\ z e. ran g)))
9		ssiun2 2589	. . . . 5 |- (g e. C -> dom g (_ U_g e. C dom g)
10		dmuni 3315	. . . . 5 |- dom U. C = U_g e. C dom g
11	9, 10	syl6ssr 2105	. . . 4 |- (g e. C -> dom g (_ dom U. C)
12	11	sseld 2064	. . 3 |- (g e. C -> (<.y, z>. e. dom g -> <.y, z>. e. dom U. C))
13	12	adantl 388	. 2 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g -> <.y, z>. e. dom U. C))
14	8, 13	sylbird 205	1 |- ((C (_ H /\ g e. C) -> ((y e. ran g /\ z e. ran g) -> <.y, z>. e. dom U. C))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462   (_ wss 2044  (/)c0 2277  <.cop 2408  U.cuni 2499  U_ciun 2562   class class class wbr 2615  omcom 3127   X. cxp 3164  dom cdm 3166  ran crn 3167  -1-1-onto->wf1o 3177   ~<_ cdom 4358

This theorem is referenced by:  infxpidmlem7 7518

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193

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