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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 6673 | . . 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 6761 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) | |
7 | 4, 5, 6 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
8 | 3, 7 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 dom cdm 5557 ∘ ccom 5561 Fun wfun 6351 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 |
This theorem is referenced by: subsaliuncllem 42647 |
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