Theorem imaexd 7929

Description: The image of a set is a set. Deduction version of imaexg 7926. (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 7926	. 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 2144  Vcvv 3450   “ cima 5653

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 5390  ax-un 7752

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 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by:  mptcnfimad  7999  ghmqusnsglem1  19500  ghmqusnsg  19502  ghmquskerlem1  19503  ghmquskerco  19504  ghmquskerlem3  19506  ghmqusker  19507  gsumfs2d  33669  algextdeglem4  34399  aks6d1c2lem4  43271  aks6d1c2  43274  aks6d1c6lem2  43315  aks6d1c6lem3  43316  aks6d1c7lem1  43324  aks6d1c7lem2  43325  sge0f1o  47486  isuspgrim0lem  49085  isubgr3stgrlem5  49162  imasubclem1  50314