Theorem fnimafnex 44633

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 6661	. . 3 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
3	1, 2	ax-mp 5	. 2 ⊢ Fun 𝐹
4		fvex 6921	. 2 ⊢ (𝐺‘𝐴) ∈ V
5		funimaexg 6648	. 2 ⊢ ((Fun 𝐹 ∧ (𝐺‘𝐴) ∈ V) → (𝐹 “ (𝐺‘𝐴)) ∈ V)
6	3, 4, 5	mp2an 705	1 ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V

This proof depends on syntax axioms:   ∈ wcel 2147  Vcvv 3454   “ cima 5658  Fun wfun 6552   Fn wfn 6553  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566

This theorem is used by: (None)