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 17145 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
5 | 4 | fveq1d 6647 | . 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋)) |
6 | eqid 2798 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | relfunc 17124 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
8 | 1st2ndbr 7723 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
9 | 7, 2, 8 | sylancr 590 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
10 | 1, 6, 9 | funcf1 17128 | . . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷)) |
11 | cofu2nd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | fvco3 6737 | . . 3 ⊢ (((1st ‘𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋 ∈ 𝐵) → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) | |
13 | 10, 11, 12 | syl2anc 587 | . 2 ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
14 | 5, 13 | eqtrd 2833 | 1 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ∘ ccom 5523 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 Basecbs 16475 Func cfunc 17116 ∘func ccofu 17118 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 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-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-ixp 8445 df-func 17120 df-cofu 17122 |
This theorem is referenced by: cofucl 17150 cofuass 17151 cofull 17196 cofth 17197 catciso 17359 1st2ndprf 17448 uncf1 17478 uncf2 17479 yonedalem21 17515 yonedalem22 17520 |
Copyright terms: Public domain | W3C validator |