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Theorem abrexexd 23207
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 4940 . . 3  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 df-mpt 4095 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32rneqi 4921 . . 3  |-  ran  (
x  e.  A  |->  B )  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
4 df-rex 2562 . . . 4  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
54abbii 2408 . . 3  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
61, 3, 53eqtr4i 2326 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
7 abrexexd.1 . . 3  |-  ( ph  ->  A  e.  _V )
8 funmpt 5306 . . . 4  |-  Fun  (
x  e.  A  |->  B )
9 eqid 2296 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
109dmmpt 5184 . . . . 5  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
11 abrexexd.0 . . . . . 6  |-  F/_ x A
1211rabexgfGS 23187 . . . . 5  |-  ( A  e.  _V  ->  { x  e.  A  |  B  e.  _V }  e.  _V )
1310, 12syl5eqel 2380 . . . 4  |-  ( A  e.  _V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 5759 . . . 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 4956 . . 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 2381 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   E.wrex 2557   {crab 2560   _Vcvv 2801   {copab 4092    e. cmpt 4093   dom cdm 4705   ran crn 4706   Fun wfun 5265
This theorem is referenced by:  esumc  23445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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