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Theorem fvcod 41783
 Description: Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
fvcod.g (𝜑 → Fun 𝐺)
fvcod.a (𝜑𝐴 ∈ dom 𝐺)
fvcod.h 𝐻 = (𝐹𝐺)
Assertion
Ref Expression
fvcod (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))

Proof of Theorem fvcod
StepHypRef Expression
1 fvcod.h . . . 4 𝐻 = (𝐹𝐺)
21fveq1i 6662 . . 3 (𝐻𝐴) = ((𝐹𝐺)‘𝐴)
32a1i 11 . 2 (𝜑 → (𝐻𝐴) = ((𝐹𝐺)‘𝐴))
4 fvcod.g . . 3 (𝜑 → Fun 𝐺)
5 fvcod.a . . 3 (𝜑𝐴 ∈ dom 𝐺)
6 fvco 6750 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
74, 5, 6syl2anc 587 . 2 (𝜑 → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
83, 7eqtrd 2859 1 (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  dom cdm 5542   ∘ ccom 5546  Fun wfun 6337  ‘cfv 6343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351 This theorem is referenced by:  subsaliuncllem  42923
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