Theorem rnexd 7866

Description: The range of a set is a set. Deduction version of rnexd 7866. (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 7853	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → ran 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3429  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  qusrn  33469