| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cofu1 | Structured version Visualization version GIF version | ||
| Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.) |
| Ref | Expression |
|---|---|
| cofuval.b | ⊢ 𝐵 = (Base‘𝐶) |
| cofuval.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
| cofuval.g | ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) |
| cofu2nd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| cofu1 | ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
| 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 | cofu1st 17928 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
| 5 | 4 | fveq1d 6908 | . 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋)) |
| 6 | eqid 2737 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
| 7 | relfunc 17907 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
| 8 | 1st2ndbr 8067 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
| 9 | 7, 2, 8 | sylancr 587 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
| 10 | 1, 6, 9 | funcf1 17911 | . . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷)) |
| 11 | cofu2nd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 12 | fvco3 7008 | . . 3 ⊢ (((1st ‘𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋 ∈ 𝐵) → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) | |
| 13 | 10, 11, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
| 14 | 5, 13 | eqtrd 2777 | 1 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ∘ ccom 5689 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 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: cofucl 17933 cofuass 17934 cofull 17981 cofth 17982 catciso 18156 1st2ndprf 18251 uncf1 18281 uncf2 18282 yonedalem21 18318 yonedalem22 18323 cofuswapf1 48994 |
| Copyright terms: Public domain | W3C validator |