Theorem imaexd 7955

Description: The image of a set is a set. Deduction version of imaexg 7952. (Contributed by Thierry Arnoux, 14-Feb-2025.)

Hypothesis

Ref	Expression
rnexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
imaexd	⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem imaexd

Step	Hyp	Ref	Expression
1		rnexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		imaexg 7952	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   “ cima 5659

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7778

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669

This theorem is used by:  mptcnfimad  8025  ghmqusnsglem1  19527  ghmqusnsg  19529  ghmquskerlem1  19530  ghmquskerco  19531  ghmquskerlem3  19533  ghmqusker  19534  gsumfs2d  33698  edompidlem  33940  algextdeglem4  34435  aks6d1c2lem4  43380  aks6d1c2  43383  aks6d1c6lem2  43424  aks6d1c6lem3  43425  aks6d1c7lem1  43433  aks6d1c7lem2  43434  sge0f1o  47607  isuspgrim0lem  49206  isubgr3stgrlem5  49283  imasubclem1  50435