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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 17934 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
5 | 4 | fveq1d 6909 | . 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋)) |
6 | eqid 2735 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | relfunc 17913 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
8 | 1st2ndbr 8066 | . . . . 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 17917 | . . 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 2775 | 1 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ∘ ccom 5693 Rel wrel 5694 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 Basecbs 17245 Func cfunc 17905 ∘func ccofu 17907 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ixp 8937 df-func 17909 df-cofu 17911 |
This theorem is referenced by: cofucl 17939 cofuass 17940 cofull 17988 cofth 17989 catciso 18165 1st2ndprf 18262 uncf1 18293 uncf2 18294 yonedalem21 18330 yonedalem22 18335 |
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