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Theorem mptrabex 5728
Description: If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
Hypothesis
Ref Expression
mptrabex.1  |-  A  e. 
_V
Assertion
Ref Expression
mptrabex  |-  ( x  e.  { y  e.  A  |  ph }  |->  B )  e.  _V
Distinct variable groups:    x, y, A    ph, x
Allowed substitution hints:    ph( y)    B( x, y)

Proof of Theorem mptrabex
StepHypRef Expression
1 mptrabex.1 . . 3  |-  A  e. 
_V
21rabex 4134 . 2  |-  { y  e.  A  |  ph }  e.  _V
32mptex 5726 1  |-  ( x  e.  { y  e.  A  |  ph }  |->  B )  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 2142   {crab 2453   _Vcvv 2731    |-> cmpt 4051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195
This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208
This theorem is referenced by:  odzval  12199
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