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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 17149 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌))) |
7 | 6 | fveq1d 6667 | . 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 2821 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
10 | relfunc 17126 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
11 | 1st2ndbr 7735 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
12 | 10, 2, 11 | sylancr 589 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
13 | 1, 8, 9, 12, 4, 5 | funcf2 17132 | . . 3 ⊢ (𝜑 → (𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1st ‘𝐹)‘𝑌))) |
14 | cofu2.y | . . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) | |
15 | fvco3 6755 | . . 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 586 | . 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 2856 | 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 1533 ∈ wcel 2110 class class class wbr 5059 ∘ ccom 5554 Rel wrel 5555 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Basecbs 16477 Hom chom 16570 Func cfunc 17118 ∘func ccofu 17120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-ixp 8456 df-func 17122 df-cofu 17124 |
This theorem is referenced by: cofucl 17152 1st2ndprf 17450 uncf2 17481 yonedalem22 17522 |
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