Theorem fnimafnex 43825

Description: The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)

Hypothesis

Ref	Expression
fnimafnex.f	⊢ 𝐹 Fn 𝐵

Assertion

Ref	Expression
fnimafnex	⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V

Proof of Theorem fnimafnex

Step	Hyp	Ref	Expression
1		fnimafnex.f	. . 3 ⊢ 𝐹 Fn 𝐵
2		fnfun 6602	. . 3 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
3	1, 2	ax-mp 5	. 2 ⊢ Fun 𝐹
4		fvex 6857	. 2 ⊢ (𝐺‘𝐴) ∈ V
5		funimaexg 6589	. 2 ⊢ ((Fun 𝐹 ∧ (𝐺‘𝐴) ∈ V) → (𝐹 “ (𝐺‘𝐴)) ∈ V)
6	3, 4, 5	mp2an 693	1 ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3442   “ cima 5637  Fun wfun 6496   Fn wfn 6497  ‘cfv 6502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-fv 6510

This theorem is referenced by: (None)