Theorem infxpidmlem5 7557

Description: Lemma for infxpidm 7565. 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 2067	. . . . 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 7556	. . . . 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 1544	. . 3 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> <.y, z>. e. (ran g X. ran g)))
6		visset 1816	. . . 4 |- z e. V
7	6	opelxp 3220	. . 3 |- (<.y, z>. e. (ran g X. ran g) <-> (y e. ran g /\ z e. ran g))
8	5, 7	syl6bb 538	. 2 |- ((C (_ H /\ g e. C) -> (<.y, z>. e. dom g <-> (y e. ran g /\ z e. ran g)))
9		ssiun2 2597	. . . . 5 |- (g e. C -> dom g (_ U_g e. C dom g)
10		dmuni 3325	. . . . 5 |- dom U. C = U_g e. C dom g
11	9, 10	syl6ssr 2111	. . . 4 |- (g e. C -> dom g (_ dom U. C)
12	11	sseld 2070	. . 3 |- (g e. C -> (<.y, z>. e. dom g -> <.y, z>. e. dom U. C))
13	12	adantl 390	. 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 958   e. wcel 960  E.wex 982  {cab 1466   (_ wss 2050  (/)c0 2283  <.cop 2415  U.cuni 2507  U_ciun 2570   class class class wbr 2624  omcom 3137   X. cxp 3174  dom cdm 3176  ran crn 3177  -1-1-onto->wf1o 3187   ~<_ cdom 4371

This theorem is referenced by:  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-rex 1653  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-uni 2508  df-iun 2572  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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