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Theorem mptexgf 7243
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)
Hypothesis
Ref Expression
mptexgf.a 𝑥𝐴
Assertion
Ref Expression
mptexgf (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Proof of Theorem mptexgf
StepHypRef Expression
1 funmpt 6603 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2736 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 6259 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 tru 1543 . . . . . . 7
542a1i 12 . . . . . 6 (𝑥𝐴 → (𝐵 ∈ V → ⊤))
65ss2rabi 4076 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
7 mptexgf.a . . . . . 6 𝑥𝐴
87rabtru 3688 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
96, 8sseqtri 4031 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
103, 9eqsstri 4029 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
11 ssexg 5322 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1210, 11mpan 690 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
13 funex 7240 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
141, 12, 13sylancr 587 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1540  wcel 2107  wnfc 2889  {crab 3435  Vcvv 3479  wss 3950  cmpt 5224  dom cdm 5684  Fun wfun 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568
This theorem is referenced by:  esumrnmpt2  34070  exrecfnlem  37381  mptexf  45248
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