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Theorem mptexgf 6757
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 6173 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2778 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 5884 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 trud 1612 . . . . . . 7 (𝐵 ∈ V → ⊤)
54rgenw 3106 . . . . . 6 𝑥𝐴 (𝐵 ∈ V → ⊤)
6 ss2rab 3899 . . . . . 6 ({𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤} ↔ ∀𝑥𝐴 (𝐵 ∈ V → ⊤))
75, 6mpbir 223 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
8 mptexgf.a . . . . . 6 𝑥𝐴
98rabtru 3569 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
107, 9sseqtri 3856 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
113, 10eqsstri 3854 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
12 ssexg 5041 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1311, 12mpan 680 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6754 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
151, 13, 14sylancr 581 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1602  wcel 2107  wnfc 2919  wral 3090  {crab 3094  Vcvv 3398  wss 3792  cmpt 4965  dom cdm 5355  Fun wfun 6129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143
This theorem is referenced by:  numclwwlk1lem2OLDOLD  27783  esumrnmpt2  30728  mptexf  40360
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