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Theorem abrexexd 23194
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 4926 . . 3  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 df-mpt 4081 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32rneqi 4907 . . 3  |-  ran  (
x  e.  A  |->  B )  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
4 df-rex 2551 . . . 4  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
54abbii 2397 . . 3  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
61, 3, 53eqtr4i 2315 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
7 abrexexd.1 . . 3  |-  ( ph  ->  A  e.  _V )
8 funmpt 5292 . . . 4  |-  Fun  (
x  e.  A  |->  B )
9 eqid 2285 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
109dmmpt 5170 . . . . 5  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
11 abrexexd.0 . . . . . 6  |-  F/_ x A
1211rabexgfGS 23173 . . . . 5  |-  ( A  e.  _V  ->  { x  e.  A  |  B  e.  _V }  e.  _V )
1310, 12syl5eqel 2369 . . . 4  |-  ( A  e.  _V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 5745 . . . 4  |-  ( ( Fun  ( x  e.  A  |->  B )  /\  dom  ( x  e.  A  |->  B )  e.  _V )  ->  ( x  e.  A  |->  B )  e. 
_V )
158, 13, 14sylancr 644 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  B )  e.  _V )
16 rnexg 4942 . . 3  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
177, 15, 163syl 18 . 2  |-  ( ph  ->  ran  ( x  e.  A  |->  B )  e. 
_V )
186, 17syl5eqelr 2370 1  |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   {cab 2271   F/_wnfc 2408   E.wrex 2546   {crab 2549   _Vcvv 2790   {copab 4078    e. cmpt 4079   dom cdm 4691   ran crn 4692   Fun wfun 5251
This theorem is referenced by:  esumc  23432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265
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