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Theorem fnimafnex 43458
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 6667 . . 3 (𝐹 Fn 𝐵 → Fun 𝐹)
31, 2ax-mp 5 . 2 Fun 𝐹
4 fvex 6918 . 2 (𝐺𝐴) ∈ V
5 funimaexg 6652 . 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 2107  Vcvv 3479  cima 5687  Fun wfun 6554   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by: (None)
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