MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptexgf Structured version   Visualization version   GIF version

Theorem mptexgf 6962
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 6362 . 2 Fun (𝑥𝐴𝐵)
2 eqid 2798 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32dmmpt 6061 . . . 4 dom (𝑥𝐴𝐵) = {𝑥𝐴𝐵 ∈ V}
4 trud 1548 . . . . . . 7 (𝐵 ∈ V → ⊤)
54rgenw 3118 . . . . . 6 𝑥𝐴 (𝐵 ∈ V → ⊤)
6 ss2rab 3998 . . . . . 6 ({𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤} ↔ ∀𝑥𝐴 (𝐵 ∈ V → ⊤))
75, 6mpbir 234 . . . . 5 {𝑥𝐴𝐵 ∈ V} ⊆ {𝑥𝐴 ∣ ⊤}
8 mptexgf.a . . . . . 6 𝑥𝐴
98rabtru 3625 . . . . 5 {𝑥𝐴 ∣ ⊤} = 𝐴
107, 9sseqtri 3951 . . . 4 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
113, 10eqsstri 3949 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
12 ssexg 5191 . . 3 ((dom (𝑥𝐴𝐵) ⊆ 𝐴𝐴𝑉) → dom (𝑥𝐴𝐵) ∈ V)
1311, 12mpan 689 . 2 (𝐴𝑉 → dom (𝑥𝐴𝐵) ∈ V)
14 funex 6959 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
151, 13, 14sylancr 590 1 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wtru 1539  wcel 2111  wnfc 2936  wral 3106  {crab 3110  Vcvv 3441  wss 3881  cmpt 5110  dom cdm 5519  Fun wfun 6318
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332
This theorem is referenced by:  esumrnmpt2  31437  exrecfnlem  34796  mptexf  41871
  Copyright terms: Public domain W3C validator