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Theorem mptexgf 6976
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 6386 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2818 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 6087 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 trud 1538 . . . . . . 7 (𝐵 ∈ V → ⊤)
54rgenw 3147 . . . . . 6 𝑥𝐴 (𝐵 ∈ V → ⊤)
6 ss2rab 4044 . . . . . 6 ({𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤} ↔ ∀𝑥𝐴 (𝐵 ∈ V → ⊤))
75, 6mpbir 232 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
8 mptexgf.a . . . . . 6 𝑥𝐴
98rabtru 3674 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
107, 9sseqtri 4000 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
113, 10eqsstri 3998 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
12 ssexg 5218 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1311, 12mpan 686 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6973 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
151, 13, 14sylancr 587 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1529  wcel 2105  wnfc 2958  wral 3135  {crab 3139  Vcvv 3492  wss 3933  cmpt 5137  dom cdm 5548  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  esumrnmpt2  31226  exrecfnlem  34542  mptexf  41383
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