Theorem imaexi 42340

Description: The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
imaexi.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
imaexi	⊢ (𝐴 “ 𝐵) ∈ V

Proof of Theorem imaexi

Step	Hyp	Ref	Expression
1		imaexi.1	. 2 ⊢ 𝐴 ∈ 𝑉
2		imaexg 7659	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 “ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3400   “ cima 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7492

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-xp 5541  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548

This theorem is referenced by:  smfpimbor1lem1  43912