Theorem fnimafnex 43702

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 6592	. . 3 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
3	1, 2	ax-mp 5	. 2 ⊢ Fun 𝐹
4		fvex 6847	. 2 ⊢ (𝐺‘𝐴) ∈ V
5		funimaexg 6579	. 2 ⊢ ((Fun 𝐹 ∧ (𝐺‘𝐴) ∈ V) → (𝐹 “ (𝐺‘𝐴)) ∈ V)
6	3, 4, 5	mp2an 692	1 ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3440   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by: (None)