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Theorem abrexexd 27178
Description: Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
abrexexd.0  |-  F/_ x A
abrexexd.1  |-  ( ph  ->  A  e.  _V )
Assertion
Ref Expression
abrexexd  |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    ph( x, y)    A( x)    B( x)

Proof of Theorem abrexexd
StepHypRef Expression
1 rnopab 5247 . . 3  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 df-mpt 4507 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32rneqi 5229 . . 3  |-  ran  (
x  e.  A  |->  B )  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
4 df-rex 2820 . . . 4  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
54abbii 2601 . . 3  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
61, 3, 53eqtr4i 2506 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
7 abrexexd.1 . . 3  |-  ( ph  ->  A  e.  _V )
8 funmpt 5624 . . . 4  |-  Fun  (
x  e.  A  |->  B )
9 eqid 2467 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
109dmmpt 5502 . . . . 5  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
11 abrexexd.0 . . . . . 6  |-  F/_ x A
1211rabexgfGS 27174 . . . . 5  |-  ( A  e.  _V  ->  { x  e.  A  |  B  e.  _V }  e.  _V )
1310, 12syl5eqel 2559 . . . 4  |-  ( A  e.  _V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 6129 . . . 4  |-  ( ( Fun  ( x  e.  A  |->  B )  /\  dom  ( x  e.  A  |->  B )  e.  _V )  ->  ( x  e.  A  |->  B )  e. 
_V )
158, 13, 14sylancr 663 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  B )  e.  _V )
16 rnexg 6717 . . 3  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
177, 15, 163syl 20 . 2  |-  ( ph  ->  ran  ( x  e.  A  |->  B )  e. 
_V )
186, 17syl5eqelr 2560 1  |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   F/_wnfc 2615   E.wrex 2815   {crab 2818   _Vcvv 3113   {copab 4504    |-> cmpt 4505   dom cdm 4999   ran crn 5000   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-8 1769  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  ax-un 6577
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:  esumc  27813
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