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Theorem mptexgf 6470
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 5914 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2620 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 5618 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 a1tru 1498 . . . . . . 7 (𝐵 ∈ V → ⊤)
54rgenw 2921 . . . . . 6 𝑥𝐴 (𝐵 ∈ V → ⊤)
6 ss2rab 3670 . . . . . 6 ({𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤} ↔ ∀𝑥𝐴 (𝐵 ∈ V → ⊤))
75, 6mpbir 221 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
8 mptexgf.a . . . . . 6 𝑥𝐴
98rabtru 3355 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
107, 9sseqtri 3629 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
113, 10eqsstri 3627 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
12 ssexg 4795 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1311, 12mpan 705 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6467 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
151, 13, 14sylancr 694 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1482  wcel 1988  wnfc 2749  wral 2909  {crab 2913  Vcvv 3195  wss 3567  cmpt 4720  dom cdm 5104  Fun wfun 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by:  esumrnmpt2  30104  mptexf  39260
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