Theorem fuco11b 49299

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 49038	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
4	3	funcrcl2 49041	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		fuco11b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
6	5	func1st2nd 49038	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
7	6	funcrcl2 49041	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6	funcrcl3 49042	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		eqidd 2730	. . . . . . 7 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
10	4, 7, 8, 9	fucoelvv 49282	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
11		1st2nd2 7986	. . . . . 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 2730	. . . . 5 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
14	4, 7, 8, 12, 13	fuco1 49283	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
15	1, 14	eqtr3d 2766	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
16	15	oveqd 7386	. 2 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹))
17		ovres 7535	. . 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 2764	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   × cxp 5629   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Catccat 17601   Func cfunc 17792   ∘_func ccofu 17794   ∘_F cfuco 49278

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-func 17796  df-cofu 17798  df-fuco 49279

This theorem is referenced by:  postcofval  49326  precofval  49329