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Theorem fnimafnex 43430
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 6669 . . 3 (𝐹 Fn 𝐵 → Fun 𝐹)
31, 2ax-mp 5 . 2 Fun 𝐹
4 fvex 6920 . 2 (𝐺𝐴) ∈ V
5 funimaexg 6654 . 2 ((Fun 𝐹 ∧ (𝐺𝐴) ∈ V) → (𝐹 “ (𝐺𝐴)) ∈ V)
63, 4, 5mp2an 692 1 (𝐹 “ (𝐺𝐴)) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by: (None)
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