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

Theorem dmfex 7630
Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dmfex ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)

Proof of Theorem dmfex
StepHypRef Expression
1 fdm 6515 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 dmexg 7602 . . . 4 (𝐹𝐶 → dom 𝐹 ∈ V)
3 eleq1 2897 . . . 4 (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
42, 3syl5ib 245 . . 3 (dom 𝐹 = 𝐴 → (𝐹𝐶𝐴 ∈ V))
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹𝐶𝐴 ∈ V))
65impcom 408 1 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  dom cdm 5548  wf 6344
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-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
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-rex 3141  df-rab 3144  df-v 3494  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-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-fn 6351  df-f 6352
This theorem is referenced by:  wemoiso  7663  fopwdom  8613  fowdom  9023  wdomfil  9475  fin23lem17  9748  fin23lem32  9754  fin23lem39  9760  enfin1ai  9794  fin1a2lem7  9816  lindfmm  20899  kelac1  39541  elefmndbas2  43972
  Copyright terms: Public domain W3C validator