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Theorem dmfex 7644
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 6525 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 dmexg 7616 . . . 4 (𝐹𝐶 → dom 𝐹 ∈ V)
3 eleq1 2903 . . . 4 (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
42, 3syl5ib 246 . . 3 (dom 𝐹 = 𝐴 → (𝐹𝐶𝐴 ∈ V))
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹𝐶𝐴 ∈ V))
65impcom 410 1 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  Vcvv 3497  dom cdm 5558  wf 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-cnv 5566  df-dm 5568  df-rn 5569  df-fn 6361  df-f 6362
This theorem is referenced by:  wemoiso  7677  fopwdom  8628  fowdom  9038  wdomfil  9490  fin23lem17  9763  fin23lem32  9769  fin23lem39  9775  enfin1ai  9809  fin1a2lem7  9831  symgbasmap  18508  lindfmm  20974  kelac1  39669
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