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Theorem abrexex 5763
Description: Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in the class expression substituted for  B, which can be thought of as  B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5745, funex 5743, fnex 5741, resfunexg 5737, and funimaexg 5329. See also abrexex2 5780. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
abrexex.1  |-  A  e. 
_V
Assertion
Ref Expression
abrexex  |-  { y  |  E. x  e.  A  y  =  B }  e.  _V
Distinct variable groups:    x, y, A    y, B
Allowed substitution hint:    B( x)

Proof of Theorem abrexex
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21rnmpt 4925 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
3 abrexex.1 . . . 4  |-  A  e. 
_V
43mptex 5746 . . 3  |-  ( x  e.  A  |->  B )  e.  _V
54rnex 4942 . 2  |-  ran  (
x  e.  A  |->  B )  e.  _V
62, 5eqeltrri 2354 1  |-  { y  |  E. x  e.  A  y  =  B }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  ab2rexex  6000  kmlem10  7785  shftfval  11565  dvdsrval  15427  cmpsublem  17126  cmpsub  17127  ptrescn  17333  heibor1lem  26533  eldiophb  26836  pointsetN  29930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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