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Theorem mptexf 41866
 Description: If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6965. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
mptexf.1 𝑥𝐴
mptexf.2 𝐴 ∈ V
Assertion
Ref Expression
mptexf (𝑥𝐴𝐵) ∈ V

Proof of Theorem mptexf
StepHypRef Expression
1 mptexf.2 . 2 𝐴 ∈ V
2 mptexf.1 . . 3 𝑥𝐴
32mptexgf 6966 . 2 (𝐴 ∈ V → (𝑥𝐴𝐵) ∈ V)
41, 3ax-mp 5 1 (𝑥𝐴𝐵) ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2112  Ⅎwnfc 2939  Vcvv 3444   ↦ cmpt 5113 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336 This theorem is referenced by:  limsupequzmpt2  42353  liminfequzmpt2  42426  smflimsuplem2  43445  smflimsuplem5  43448
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