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Theorem abrexex 6540
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 5934, funex 5932, fnex 5931, resfunexg 5930, and funimaexg 5483. See also abrexex2 6547. (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 2433 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21rnmpt 5072 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
3 abrexex.1 . . . 4  |-  A  e. 
_V
43mptex 5935 . . 3  |-  ( x  e.  A  |->  B )  e.  _V
54rnex 6501 . 2  |-  ran  (
x  e.  A  |->  B )  e.  _V
62, 5eqeltrri 2504 1  |-  { y  |  E. x  e.  A  y  =  B }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   {cab 2419   E.wrex 2706   _Vcvv 2962    e. cmpt 4338   ran crn 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414
This theorem is referenced by:  ab2rexex  6557  kmlem10  8316  shftfval  12543  dvdsrval  16671  cmpsublem  18844  cmpsub  18845  ptrescn  19054  heibor1lem  28552  eldiophb  28940  pointsetN  32979
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