Theorem rnexd 7931

Description: The range of a set is a set. Deduction version of rnexd 7931. (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 7918	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3	1, 2	syl 18	1 ⊢ (𝜑 → ran 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Vcvv 3450  ran crn 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  qusrn  34010