Theorem opabex 3595

Description: Existence of a function expressed as class of ordered pairs.

Hypotheses

Ref	Expression
opabex.1	|- A e. V
opabex.2	|- (x e. A -> E*yph)

Assertion

Ref	Expression
opabex	|- {<.x, y>. | (x e. A /\ ph)} e. V

Distinct variable group:   x,y,A

Proof of Theorem opabex

Step	Hyp	Ref	Expression
1		funopab 3534	. . 3 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
2		opabex.2	. . . 4 |- (x e. A -> E*yph)
3		moanimv 1422	. . . 4 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
4	2, 3	mpbir 190	. . 3 |- E*y(x e. A /\ ph)
5	1, 4	mpgbir 985	. 2 |- Fun {<.x, y>. | (x e. A /\ ph)}
6		opabex.1	. . 3 |- A e. V
7		dmopabss 3310	. . 3 |- dom {<.x, y>. | (x e. A /\ ph)} (_ A
8	6, 7	ssexi 2710	. 2 |- dom {<.x, y>. | (x e. A /\ ph)} e. V
9		funex 3594	. 2 |- ((Fun {<.x, y>. | (x e. A /\ ph)} /\ dom {<.x, y>. | (x e. A /\ ph)} e. V) -> {<.x, y>. | (x e. A /\ ph)} e. V)
10	5, 8, 9	mp2an 695	1 |- {<.x, y>. | (x e. A /\ ph)} e. V

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  E*wmo 1374  Vcvv 1802  {copab 2656  dom cdm 3160  Fun wfun 3166

This theorem is referenced by:  opabex2 3596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183

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