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Theorem abrexex 5775
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 5418, funex 5416, fnex 5415, resfunexg 5414, and funimaexg 5014. See also abrexex2 5782. (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 2082 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
21rnmpt 4610 . 2  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
3 abrexex.1 . . . 4  |-  A  e. 
_V
43mptex 5419 . . 3  |-  ( x  e.  A  |->  B )  e.  _V
54rnex 4627 . 2  |-  ran  (
x  e.  A  |->  B )  e.  _V
62, 5eqeltrri 2153 1  |-  { y  |  E. x  e.  A  y  =  B }  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   {cab 2068   E.wrex 2350   _Vcvv 2602    |-> cmpt 3847   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940
This theorem is referenced by:  ab2rexex  5789  shftfval  9847
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