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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvcod | Structured version Visualization version GIF version |
Description: Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fvcod.g | ⊢ (𝜑 → Fun 𝐺) |
fvcod.a | ⊢ (𝜑 → 𝐴 ∈ dom 𝐺) |
fvcod.h | ⊢ 𝐻 = (𝐹 ∘ 𝐺) |
Ref | Expression |
---|---|
fvcod | ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvcod.h | . . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐺) | |
2 | 1 | fveq1i 6908 | . . 3 ⊢ (𝐻‘𝐴) = ((𝐹 ∘ 𝐺)‘𝐴) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝐻‘𝐴) = ((𝐹 ∘ 𝐺)‘𝐴)) |
4 | fvcod.g | . . 3 ⊢ (𝜑 → Fun 𝐺) | |
5 | fvcod.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝐺) | |
6 | fvco 7007 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) | |
7 | 4, 5, 6 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
8 | 3, 7 | eqtrd 2775 | 1 ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
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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