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Theorem fvcod 45170
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 6908 . . 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 2775 1 (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  dom cdm 5689  ccom 5693  Fun wfun 6557  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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  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:  subsaliuncllem  46313
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