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Theorem mptrabex 6631
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 ∈ V
Assertion
Ref Expression
mptrabex (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem mptrabex
StepHypRef Expression
1 mptrabex.1 . . 3 𝐴 ∈ V
21rabex 4946 . 2 {𝑦𝐴𝜑} ∈ V
32mptex 6629 1 (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  {crab 3065  Vcvv 3351  cmpt 4863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  odzval  15702  pmtrfval  18076  dmdprd  18604  dprdval  18609  psrlidm  19617  psrass23l  19622  psrass23  19624  mplsubrg  19654  mplmonmul  19678  mplbas2  19684  fusgrfis  26444  wlknwwlksnbij2OLD  27026  wlkwwlkbij2OLD  27033  wlksnwwlknvbij  27051  wlksnwwlknvbijOLD  27052  clwwlkvbij  27288  clwwlkvbijOLDOLD  27289  clwwlkvbijOLD  27290  sitgval  30731  fwddifnval  32604  diafval  36837  dicfval  36981
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