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Theorem mptrabex 6983
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 5232 . 2 {𝑦𝐴𝜑} ∈ V
32mptex 6981 1 (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {crab 3147  Vcvv 3500  cmpt 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360
This theorem is referenced by:  odzval  16118  pmtrfval  18498  dmdprd  19040  dprdval  19045  psrlidm  20102  psrass23l  20107  psrass23  20109  mplsubrg  20139  mplmonmul  20163  mplbas2  20169  fusgrfis  27026  wlksnwwlknvbij  27601  clwwlkvbij  27806  sitgval  31476  fwddifnval  33508  diafval  38034  dicfval  38178
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