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Theorem dmfex 7359
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 6264 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 dmexg 7331 . . . 4 (𝐹𝐶 → dom 𝐹 ∈ V)
3 eleq1 2866 . . . 4 (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
42, 3syl5ib 236 . . 3 (dom 𝐹 = 𝐴 → (𝐹𝐶𝐴 ∈ V))
51, 4syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹𝐶𝐴 ∈ V))
65impcom 397 1 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  dom cdm 5312  wf 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-cnv 5320  df-dm 5322  df-rn 5323  df-fn 6104  df-f 6105
This theorem is referenced by:  wemoiso  7386  fopwdom  8310  fowdom  8718  wdomfil  9170  fin23lem17  9448  fin23lem32  9454  fin23lem39  9460  enfin1ai  9494  fin1a2lem7  9516  lindfmm  20491  kelac1  38418
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