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Theorem abrexexd 27979
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 5099 . . 3  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 df-mpt 4486 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32rneqi 5081 . . 3  |-  ran  (
x  e.  A  |->  B )  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
4 df-rex 2788 . . . 4  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
54abbii 2563 . . 3  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
61, 3, 53eqtr4i 2468 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
7 abrexexd.1 . . 3  |-  ( ph  ->  A  e.  _V )
8 funmpt 5637 . . . 4  |-  Fun  (
x  e.  A  |->  B )
9 eqid 2429 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
109dmmpt 5350 . . . . 5  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
11 abrexexd.0 . . . . . 6  |-  F/_ x A
1211rabexgfGS 27973 . . . . 5  |-  ( A  e.  _V  ->  { x  e.  A  |  B  e.  _V }  e.  _V )
1310, 12syl5eqel 2521 . . . 4  |-  ( A  e.  _V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 6148 . . . 4  |-  ( ( Fun  ( x  e.  A  |->  B )  /\  dom  ( x  e.  A  |->  B )  e.  _V )  ->  ( x  e.  A  |->  B )  e. 
_V )
158, 13, 14sylancr 667 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  B )  e.  _V )
16 rnexg 6739 . . 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 2522 1  |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   F/_wnfc 2577   E.wrex 2783   {crab 2786   _Vcvv 3087   {copab 4483    |-> cmpt 4484   dom cdm 4854   ran crn 4855   Fun wfun 5595
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by:  esumc  28711
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