Theorem fndmexd 7927

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 6674	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐷)
3		fndmexd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4	3	dmexd 7926	. 2 ⊢ (𝜑 → dom 𝐹 ∈ V)
5	2, 4	eqeltrrd 2840	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478  dom cdm 5689   Fn wfn 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566

This theorem is referenced by:  fndmexb  7929  fsetdmprc0  8894  finnzfsuppd  9411  psrbagfsupp  21957  psrbaglecl  21961  psrbagaddcl  21962  psrbagcon  21963  psrbagleadd1  21966  psrbagconf1o  21967  gsumbagdiaglem  21968  psrass1lem  21970  psrbagev1  22119  psrbagev2  22120  tdeglem1  26112  tdeglem3  26113  tdeglem4  26114  gsumhashmul  33047  mhphf  42584