Theorem rnexd 7916

Description: The range of a set is a set. Deduction version of rnexd 7916. (Contributed by Thierry Arnoux, 14-Feb-2025.)

Hypothesis

Ref	Expression
rnexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
rnexd	⊢ (𝜑 → ran 𝐴 ∈ V)

Proof of Theorem rnexd

Step	Hyp	Ref	Expression
1		rnexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		rnexg 7903	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → ran 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464  ran crn 5660

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  qusrn  33429