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Theorem abrexexg 6760
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 2443 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21rnmpt 5238 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
3 mptexg 6127 . . 3  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 6717 . . 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 2536 1  |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   {cab 2428   E.wrex 2794   _Vcvv 3095    |-> cmpt 4495   ran crn 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586
This theorem is referenced by:  iunexg  6761  qsexg  7371  wdomd  8010  cardiun  8366  rankcf  9158  sigaclci  28005  hbtlem1  31047  hbtlem7  31049
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