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Theorem fndmexd 40834
Description: If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
Hypotheses
Ref Expression
fndmexd.1 (𝜑𝐹𝑉)
fndmexd.2 (𝜑𝐹 Fn 𝐷)
Assertion
Ref Expression
fndmexd (𝜑𝐷 ∈ V)

Proof of Theorem fndmexd
StepHypRef Expression
1 fndmexd.2 . . 3 (𝜑𝐹 Fn 𝐷)
21fndmd 6445 . 2 (𝜑 → dom 𝐹 = 𝐷)
3 fndmexd.1 . . 3 (𝜑𝐹𝑉)
43dmexd 7610 . 2 (𝜑 → dom 𝐹 ∈ V)
52, 4eqeltrrd 2917 1 (𝜑𝐷 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3480  dom cdm 5542   Fn wfn 6338
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553  df-fn 6346
This theorem is referenced by:  finnzfsuppd  40835
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