Theorem fuco11b 49498

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 49237	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
4	3	funcrcl2 49240	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		fuco11b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
6	5	func1st2nd 49237	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
7	6	funcrcl2 49240	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6	funcrcl3 49241	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		eqidd 2734	. . . . . . 7 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
10	4, 7, 8, 9	fucoelvv 49481	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
11		1st2nd2 7969	. . . . . 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 2734	. . . . 5 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
14	4, 7, 8, 12, 13	fuco1 49482	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
15	1, 14	eqtr3d 2770	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
16	15	oveqd 7372	. 2 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹))
17		ovres 7521	. . 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 584	. 2 ⊢ (𝜑 → (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹) = (𝐺 ∘func 𝐹))
19	16, 18	eqtrd 2768	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4583   × cxp 5619   ↾ cres 5623  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Catccat 17578   Func cfunc 17769   ∘_func ccofu 17771   ∘_F cfuco 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-func 17773  df-cofu 17775  df-fuco 49478

This theorem is referenced by:  postcofval  49525  precofval  49528