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Theorem mptexgf 28627
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 5725 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2514 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 5437 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 a1tru 1490 . . . . . . 7 (𝐵 ∈ V → ⊤)
54rgenw 2812 . . . . . 6 𝑥𝐴 (𝐵 ∈ V → ⊤)
6 ss2rab 3545 . . . . . 6 ({𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤} ↔ ∀𝑥𝐴 (𝐵 ∈ V → ⊤))
75, 6mpbir 219 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
8 mptexgf.a . . . . . 6 𝑥𝐴
98rabtru 28541 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
107, 9sseqtri 3504 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
113, 10eqsstri 3502 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
12 ssexg 4631 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1311, 12mpan 701 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6263 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
151, 13, 14sylancr 693 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1475  wcel 1938  wnfc 2642  wral 2800  {crab 2804  Vcvv 3077  wss 3444  cmpt 4541  dom cdm 4932  Fun wfun 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697
This theorem is referenced by:  esumrnmpt2  29283
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