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Theorem opabex 6130
Description: Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
Hypotheses
Ref Expression
opabex.1  |-  A  e. 
_V
opabex.2  |-  ( x  e.  A  ->  E* y ph )
Assertion
Ref Expression
opabex  |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabex
StepHypRef Expression
1 funopab 5621 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
2 opabex.2 . . . 4  |-  ( x  e.  A  ->  E* y ph )
3 moanimv 2356 . . . 4  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
42, 3mpbir 209 . . 3  |-  E* y
( x  e.  A  /\  ph )
51, 4mpgbir 1605 . 2  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
6 opabex.1 . . 3  |-  A  e. 
_V
7 dmopabss 5214 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
86, 7ssexi 4592 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
9 funex 6129 . 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 )
105, 8, 9mp2an 672 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   E*wmo 2276   _Vcvv 3113   {copab 4504   dom cdm 4999   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by: (None)
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