Theorem fuco11b 49827

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		fuco11b.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3	2	func1st2nd 49566	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
4	3	funcrcl2 49569	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		fuco11b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
6	5	func1st2nd 49566	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
7	6	funcrcl2 49569	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6	funcrcl3 49570	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		eqidd 2740	. . . . . . 7 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
10	4, 7, 8, 9	fucoelvv 49810	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
11		1st2nd2 7970	. . . . . 6 ⊢ ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
12	10, 11	syl 17	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)), (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟩)
13		eqidd 2740	. . . . 5 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
14	4, 7, 8, 12, 13	fuco1 49811	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
15	1, 14	eqtr3d 2776	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
16	15	oveqd 7373	. 2 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹))
17		ovres 7522	. . 3 ⊢ ((𝐺 ∈ (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹) = (𝐺 ∘func 𝐹))
18	5, 2, 17	syl2anc 590	. 2 ⊢ (𝜑 → (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹) = (𝐺 ∘func 𝐹))
19	16, 18	eqtrd 2774	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   × cxp 5616   ↾ cres 5620  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Catccat 17621   Func cfunc 17812   ∘_func ccofu 17814   ∘_F cfuco 49806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-func 17816  df-cofu 17818  df-fuco 49807

This theorem is referenced by:  postcofval  49854  precofval  49857