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Theorem mptexgf 7223
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 6586 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2732 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 6239 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 tru 1545 . . . . . . 7
542a1i 12 . . . . . 6 (𝑥𝐴 → (𝐵 ∈ V → ⊤))
65ss2rabi 4074 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
7 mptexgf.a . . . . . 6 𝑥𝐴
87rabtru 3680 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
96, 8sseqtri 4018 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
103, 9eqsstri 4016 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
11 ssexg 5323 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1210, 11mpan 688 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
13 funex 7220 . 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 1542  wcel 2106  wnfc 2883  {crab 3432  Vcvv 3474  wss 3948  cmpt 5231  dom cdm 5676  Fun wfun 6537
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  esumrnmpt2  33061  exrecfnlem  36255  mptexf  43930
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