Theorem rnexd 7918

Description: The range of a set is a set. Deduction version of rnexd 7918. (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 7905	. 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 2146  Vcvv 3457  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  qusrn  33784