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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 16366 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
5 | 4 | fveq1d 6105 | . 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋)) |
6 | eqid 2610 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | relfunc 16345 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
8 | 1st2ndbr 7108 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
9 | 7, 2, 8 | sylancr 694 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
10 | 1, 6, 9 | funcf1 16349 | . . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷)) |
11 | cofu2nd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | fvco3 6185 | . . 3 ⊢ (((1st ‘𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋 ∈ 𝐵) → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) | |
13 | 10, 11, 12 | syl2anc 691 | . 2 ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
14 | 5, 13 | eqtrd 2644 | 1 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ∘ ccom 5042 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Func cfunc 16337 ∘func ccofu 16339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-ixp 7795 df-func 16341 df-cofu 16343 |
This theorem is referenced by: cofucl 16371 cofuass 16372 cofull 16417 cofth 16418 catciso 16580 1st2ndprf 16669 uncf1 16699 uncf2 16700 yonedalem21 16736 yonedalem22 16741 |
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