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Theorem opabex 6150
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 5622 . . 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 2380 . . . 4  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
42, 3mpbir 214 . . 3  |-  E* y
( x  e.  A  /\  ph )
51, 4mpgbir 1681 . 2  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
6 opabex.1 . . 3  |-  A  e. 
_V
7 dmopabss 5052 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
86, 7ssexi 4541 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
9 funex 6149 . 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 686 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904   E*wmo 2320   _Vcvv 3031   {copab 4453   dom cdm 4839   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597
This theorem is referenced by: (None)
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