Theorem imaexd 7922

Description: The image of a set is a set. Deduction version of imaexg 7919. (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 7919	. 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 2146  Vcvv 3458   “ cima 5669

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 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by:  mptcnfimad  7992  ghmqusnsglem1  19381  ghmqusnsg  19383  ghmquskerlem1  19384  ghmquskerco  19385  ghmquskerlem3  19387  ghmqusker  19388  gsumfs2d  33412  algextdeglem4  34141  aks6d1c2lem4  42935  aks6d1c2  42938  aks6d1c6lem2  42979  aks6d1c6lem3  42980  aks6d1c7lem1  42988  aks6d1c7lem2  42989  sge0f1o  47137  isuspgrim0lem  48699  isubgr3stgrlem5  48776  imasubclem1  49923