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Mirrors > Home > MPE Home > Th. List > cofu2 | Structured version Visualization version GIF version |
Description: Value of the morphism 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) |
cofu2nd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
cofu2.h | ⊢ 𝐻 = (Hom ‘𝐶) |
cofu2.y | ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
cofu2 | ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅))) |
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 | cofu2nd.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | cofu2nd.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | cofu2nd 17143 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))) |
7 | 6 | fveq1d 6665 | . 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅)) |
8 | cofu2.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | eqid 2818 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | relfunc 17120 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
11 | 1st2ndbr 7730 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
12 | 10, 2, 11 | sylancr 587 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
13 | 1, 8, 9, 12, 4, 5 | funcf2 17126 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌))) |
14 | cofu2.y | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) | |
15 | fvco3 6753 | . . 3 ⊢ (((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌)) ∧ 𝑅 ∈ (𝑋𝐻𝑌)) → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅))) | |
16 | 13, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → (((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅))) |
17 | 7, 16 | eqtrd 2853 | 1 ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ∘ ccom 5552 Rel wrel 5553 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 Basecbs 16471 Hom chom 16564 Func cfunc 17112 ∘func ccofu 17114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 df-ixp 8450 df-func 17116 df-cofu 17118 |
This theorem is referenced by: cofucl 17146 1st2ndprf 17444 uncf2 17475 yonedalem22 17516 |
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