Theorem fndmexd 7851

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 6597	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐷)
3		fndmexd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4	3	dmexd 7850	. 2 ⊢ (𝜑 → dom 𝐹 ∈ V)
5	2, 4	eqeltrrd 2841	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3432  dom cdm 5625   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-cnv 5633  df-dm 5635  df-rn 5636  df-fn 6495

This theorem is referenced by:  fndmexb  7853  fsetdmprc0  8799  finnzfsuppd  9283  psrbagfsupp  21901  psrbaglecl  21905  psrbagaddcl  21906  psrbagcon  21907  psrbagleadd1  21910  psrbagconf1o  21911  gsumbagdiaglem  21913  psrass1lem  21915  psrbagev1  22060  psrbagev2  22061  tdeglem1  26048  tdeglem3  26049  tdeglem4  26050  gsumhashmul  33155  mhphf  43048