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Theorem abrexexg 5986
Description: Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in  B. The antecedent assures us that  A is a set. (Contributed by NM, 3-Nov-2003.)
Assertion
Ref Expression
abrexexg  |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Distinct variable groups:    x, y, A    y, B
Allowed substitution hints:    B( x)    V( x, y)

Proof of Theorem abrexexg
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21rnmpt 5118 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
3 mptexg 5967 . . 3  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 5133 . . 3  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
53, 4syl 16 . 2  |-  ( A  e.  V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
62, 5syl5eqelr 2523 1  |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  iunexg  5989  qsexg  6964  wdomd  7551  cardiun  7871  rankcf  8654  sigaclci  24517  hbtlem1  27306  hbtlem7  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
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