Theorem fndmexd 7846

Description: If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)

Hypotheses

Ref	Expression
fndmexd.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
fndmexd.2	⊢ (𝜑 → 𝐹 Fn 𝐷)

Assertion

Ref	Expression
fndmexd	⊢ (𝜑 → 𝐷 ∈ V)

Proof of Theorem fndmexd

Step	Hyp	Ref	Expression
1		fndmexd.2	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷)
2	1	fndmd 6595	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐷)
3		fndmexd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4	3	dmexd 7845	. 2 ⊢ (𝜑 → dom 𝐹 ∈ V)
5	2, 4	eqeltrrd 2838	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430  dom cdm 5622   Fn wfn 6485

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-cnv 5630  df-dm 5632  df-rn 5633  df-fn 6493

This theorem is referenced by:  fndmexb  7848  fsetdmprc0  8793  finnzfsuppd  9277  psrbagfsupp  21876  psrbaglecl  21880  psrbagaddcl  21881  psrbagcon  21882  psrbagleadd1  21885  psrbagconf1o  21886  gsumbagdiaglem  21887  psrass1lem  21889  psrbagev1  22033  psrbagev2  22034  tdeglem1  26004  tdeglem3  26005  tdeglem4  26006  gsumhashmul  33133  mhphf  43029