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Theorem opabex 6140
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 5625 . . 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 2325 . . . 4  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
42, 3mpbir 212 . . 3  |-  E* y
( x  e.  A  /\  ph )
51, 4mpgbir 1669 . 2  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
6 opabex.1 . . 3  |-  A  e. 
_V
7 dmopabss 5057 . . 3  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
86, 7ssexi 4561 . 2  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
9 funex 6139 . 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 676 1  |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1867   E*wmo 2264   _Vcvv 3078   {copab 4474   dom cdm 4845   Fun wfun 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600
This theorem is referenced by: (None)
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