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Theorem fndmexd 7844
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 6608 . 2 (𝜑 → dom 𝐹 = 𝐷)
3 fndmexd.1 . . 3 (𝜑𝐹𝑉)
43dmexd 7843 . 2 (𝜑 → dom 𝐹 ∈ V)
52, 4eqeltrrd 2835 1 (𝜑𝐷 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444  dom cdm 5634   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-fn 6500
This theorem is referenced by:  fndmexb  7846  fsetdmprc0  8796  psrbagfsupp  21338  psrbaglecl  21344  psrbagaddcl  21346  psrbagcon  21348  psrbagconf1o  21354  gsumbagdiaglem  21359  psrass1lem  21361  psrbagev1  21501  psrbagev2  21503  tdeglem1  25436  tdeglem3  25438  tdeglem4  25440  gsumhashmul  31947  finnzfsuppd  42570
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