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| Mirrors > Home > MPE Home > Th. List > cofu1st | Structured version Visualization version GIF version | ||
| Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| Ref | Expression |
|---|---|
| cofuval.b | ⊢ 𝐵 = (Base‘𝐶) |
| cofuval.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
| cofuval.g | ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) |
| Ref | Expression |
|---|---|
| cofu1st | ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cofuval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | cofuval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
| 3 | cofuval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) | |
| 4 | 1, 2, 3 | cofuval 17927 | . . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉) |
| 5 | 4 | fveq2d 6910 | . 2 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉)) |
| 6 | fvex 6919 | . . . 4 ⊢ (1st ‘𝐺) ∈ V | |
| 7 | fvex 6919 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
| 8 | 6, 7 | coex 7952 | . . 3 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V |
| 9 | 1 | fvexi 6920 | . . . 4 ⊢ 𝐵 ∈ V |
| 10 | 9, 9 | mpoex 8104 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V |
| 11 | 8, 10 | op1st 8022 | . 2 ⊢ (1st ‘〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉) = ((1st ‘𝐺) ∘ (1st ‘𝐹)) |
| 12 | 5, 11 | eqtrdi 2793 | 1 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 Func cfunc 17899 ∘func ccofu 17901 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 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-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ixp 8938 df-func 17903 df-cofu 17905 |
| This theorem is referenced by: cofu1 17929 cofucl 17933 cofuass 17934 catciso 18156 |
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