Theorem fndmexd 7860

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 6605	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐷)
3		fndmexd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4	3	dmexd 7859	. 2 ⊢ (𝜑 → dom 𝐹 ∈ V)
5	2, 4	eqeltrrd 2829	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3444  dom cdm 5631   Fn wfn 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-cnv 5639  df-dm 5641  df-rn 5642  df-fn 6502

This theorem is referenced by:  fndmexb  7862  fsetdmprc0  8805  finnzfsuppd  9300  psrbagfsupp  21861  psrbaglecl  21865  psrbagaddcl  21866  psrbagcon  21867  psrbagleadd1  21870  psrbagconf1o  21871  gsumbagdiaglem  21872  psrass1lem  21874  psrbagev1  22017  psrbagev2  22018  tdeglem1  25996  tdeglem3  25997  tdeglem4  25998  gsumhashmul  33044  mhphf  42578