Theorem fndmexd 7912

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 6660	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐷)
3		fndmexd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
4	3	dmexd 7911	. 2 ⊢ (𝜑 → dom 𝐹 ∈ V)
5	2, 4	eqeltrrd 2826	1 ⊢ (𝜑 → 𝐷 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3461  dom cdm 5678   Fn wfn 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5686  df-dm 5688  df-rn 5689  df-fn 6552

This theorem is referenced by:  fndmexb  7914  fsetdmprc0  8874  psrbagfsupp  21870  psrbaglecl  21876  psrbagaddcl  21878  psrbagcon  21880  psrbagleadd1  21886  psrbagconf1o  21887  gsumbagdiaglem  21892  psrass1lem  21894  psrbagev1  22043  psrbagev2  22045  tdeglem1  26035  tdeglem3  26037  tdeglem4  26039  gsumhashmul  32860  finnzfsuppd  43778