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Theorem fvcod 45232
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 6907 . . 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 7007 . . 3 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
74, 5, 6syl2anc 584 . 2 (𝜑 → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
83, 7eqtrd 2777 1 (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  dom cdm 5685  ccom 5689  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  subsaliuncllem  46372
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