Theorem fuco11b 49005

Description: The object part of the functor composition bifunctor maps two functors to their composition. (Contributed by Zhi Wang, 11-Oct-2025.)

Hypotheses

Ref	Expression
fuco11b.o	⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)
fuco11b.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
fuco11b.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
fuco11b	⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Proof of Theorem fuco11b

Step	Hyp	Ref	Expression
1		fuco11b.o	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)
2		relfunc 17903	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
3		fuco11b.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		1st2ndbr 8063	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
5	2, 3, 4	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
6	5	funcrcl2 48885	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		relfunc 17903	. . . . . . 7 ⊢ Rel (𝐷 Func 𝐸)
8		fuco11b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
9		1st2ndbr 8063	. . . . . . 7 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
10	7, 8, 9	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
11	10	funcrcl2 48885	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	10	funcrcl3 48886	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		eqidd 2737	. . . . . . 7 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
14	6, 11, 12, 13	fucoelvv 48988	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
15		1st2nd2 8049	. . . . . 6 ⊢ ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
17		eqidd 2737	. . . . 5 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
18	6, 11, 12, 16, 17	fuco1 48989	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
19	1, 18	eqtr3d 2778	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
20	19	oveqd 7446	. 2 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹))
21		ovres 7596	. . 3 ⊢ ((𝐺 ∈ (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹) = (𝐺 ∘func 𝐹))
22	8, 3, 21	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹) = (𝐺 ∘func 𝐹))
23	20, 22	eqtrd 2776	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3479  ⟨cop 4630   class class class wbr 5141   × cxp 5681   ↾ cres 5685  Rel wrel 5688  ‘cfv 6559  (class class class)co 7429  1^st c1st 8008  2^nd c2nd 8009  Catccat 17703   Func cfunc 17895   ∘_func ccofu 17897   ∘_F cfuco 48984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-func 17899  df-cofu 17901  df-fuco 48985

This theorem is referenced by:  postcofval  49032  precofval  49035