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Mirrors > Home > MPE Home > Th. List > cofu1st | Structured version Visualization version GIF version |
Description: Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
cofuval.b | ⊢ 𝐵 = (Base‘𝐶) |
cofuval.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
cofuval.g | ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) |
Ref | Expression |
---|---|
cofu1st | ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
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 | cofuval 17833 | . . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉) |
5 | 4 | fveq2d 6886 | . 2 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = (1st ‘〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉)) |
6 | fvex 6895 | . . . 4 ⊢ (1st ‘𝐺) ∈ V | |
7 | fvex 6895 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
8 | 6, 7 | coex 7915 | . . 3 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V |
9 | 1 | fvexi 6896 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9, 9 | mpoex 8060 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V |
11 | 8, 10 | op1st 7977 | . 2 ⊢ (1st ‘〈((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))〉) = ((1st ‘𝐺) ∘ (1st ‘𝐹)) |
12 | 5, 11 | eqtrdi 2780 | 1 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 〈cop 4627 ∘ ccom 5671 ‘cfv 6534 (class class class)co 7402 ∈ cmpo 7404 1st c1st 7967 2nd c2nd 7968 Basecbs 17145 Func cfunc 17805 ∘func ccofu 17807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 df-ixp 8889 df-func 17809 df-cofu 17811 |
This theorem is referenced by: cofu1 17835 cofucl 17839 cofuass 17840 catciso 18065 |
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