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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 16895 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹))) |
5 | 4 | fveq1d 6435 | . 2 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋)) |
6 | eqid 2825 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
7 | relfunc 16874 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
8 | 1st2ndbr 7479 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
9 | 7, 2, 8 | sylancr 583 | . . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
10 | 1, 6, 9 | funcf1 16878 | . . 3 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶(Base‘𝐷)) |
11 | cofu2nd.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | fvco3 6522 | . . 3 ⊢ (((1st ‘𝐹):𝐵⟶(Base‘𝐷) ∧ 𝑋 ∈ 𝐵) → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) | |
13 | 10, 11, 12 | syl2anc 581 | . 2 ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
14 | 5, 13 | eqtrd 2861 | 1 ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4873 ∘ ccom 5346 Rel wrel 5347 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 1st c1st 7426 2nd c2nd 7427 Basecbs 16222 Func cfunc 16866 ∘func ccofu 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 df-ixp 8176 df-func 16870 df-cofu 16872 |
This theorem is referenced by: cofucl 16900 cofuass 16901 cofull 16946 cofth 16947 catciso 17109 1st2ndprf 17199 uncf1 17229 uncf2 17230 yonedalem21 17266 yonedalem22 17271 |
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