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Theorem fnimafnex 41719
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 “ (𝐺𝐴)) ∈ V

Proof of Theorem fnimafnex
StepHypRef Expression
1 fnimafnex.f . . 3 𝐹 Fn 𝐵
2 fnfun 6603 . . 3 (𝐹 Fn 𝐵 → Fun 𝐹)
31, 2ax-mp 5 . 2 Fun 𝐹
4 fvex 6856 . 2 (𝐺𝐴) ∈ V
5 funimaexg 6588 . 2 ((Fun 𝐹 ∧ (𝐺𝐴) ∈ V) → (𝐹 “ (𝐺𝐴)) ∈ V)
63, 4, 5mp2an 691 1 (𝐹 “ (𝐺𝐴)) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3446  cima 5637  Fun wfun 6491   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  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 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by: (None)
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