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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 6646 | . . 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 6736 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) | |
7 | 4, 5, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
8 | 3, 7 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 dom cdm 5519 ∘ ccom 5523 Fun wfun 6318 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: subsaliuncllem 42997 |
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