ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mptrabex Unicode version

Theorem mptrabex 5721
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 4131 . 2  |-  { y  e.  A  |  ph }  e.  _V
32mptex 5719 1  |-  ( x  e.  { y  e.  A  |  ph }  |->  B )  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 2141   {crab 2452   _Vcvv 2730    |-> cmpt 4048
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204
This theorem is referenced by:  odzval  12182
  Copyright terms: Public domain W3C validator