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Theorem mptrabex 6717
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 5007 . 2 {𝑦𝐴𝜑} ∈ V
32mptex 6715 1 (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  {crab 3093  Vcvv 3385  cmpt 4922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109
This theorem is referenced by:  odzval  15829  pmtrfval  18182  dmdprd  18713  dprdval  18718  psrlidm  19726  psrass23l  19731  psrass23  19733  mplsubrg  19763  mplmonmul  19787  mplbas2  19793  fusgrfis  26564  wlknwwlksnbij2OLD  27151  wlkwwlkbij2OLD  27158  wlksnwwlknvbij  27188  wlksnwwlknvbijOLD  27189  clwwlkvbij  27453  clwwlkvbijOLD  27454  clwwlkvbijOLDOLD  27455  sitgval  30910  fwddifnval  32783  diafval  37052  dicfval  37196
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