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Theorem abrexexd 23982
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 5107 . . 3  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 df-mpt 4260 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32rneqi 5088 . . 3  |-  ran  (
x  e.  A  |->  B )  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
4 df-rex 2703 . . . 4  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
54abbii 2547 . . 3  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
61, 3, 53eqtr4i 2465 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
7 abrexexd.1 . . 3  |-  ( ph  ->  A  e.  _V )
8 funmpt 5481 . . . 4  |-  Fun  (
x  e.  A  |->  B )
9 eqid 2435 . . . . . 6  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
109dmmpt 5357 . . . . 5  |-  dom  (
x  e.  A  |->  B )  =  { x  e.  A  |  B  e.  _V }
11 abrexexd.0 . . . . . 6  |-  F/_ x A
1211rabexgfGS 23979 . . . . 5  |-  ( A  e.  _V  ->  { x  e.  A  |  B  e.  _V }  e.  _V )
1310, 12syl5eqel 2519 . . . 4  |-  ( A  e.  _V  ->  dom  ( x  e.  A  |->  B )  e.  _V )
14 funex 5955 . . . 4  |-  ( ( Fun  ( x  e.  A  |->  B )  /\  dom  ( x  e.  A  |->  B )  e.  _V )  ->  ( x  e.  A  |->  B )  e. 
_V )
158, 13, 14sylancr 645 . . 3  |-  ( A  e.  _V  ->  (
x  e.  A  |->  B )  e.  _V )
16 rnexg 5123 . . 3  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
177, 15, 163syl 19 . 2  |-  ( ph  ->  ran  ( x  e.  A  |->  B )  e. 
_V )
186, 17syl5eqelr 2520 1  |-  ( ph  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558   E.wrex 2698   {crab 2701   _Vcvv 2948   {copab 4257    e. cmpt 4258   dom cdm 4870   ran crn 4871   Fun wfun 5440
This theorem is referenced by:  esumc  24438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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