Theorem fuco11b 49369

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 49108	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
4	3	funcrcl2 49111	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		fuco11b.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
6	5	func1st2nd 49108	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐺)(𝐷 Func 𝐸)(2nd ‘𝐺))
7	6	funcrcl2 49111	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	6	funcrcl3 49112	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
9		eqidd 2732	. . . . . . 7 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = (⟨𝐶, 𝐷⟩ ∘F 𝐸))
10	4, 7, 8, 9	fucoelvv 49352	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (V × V))
11		1st2nd2 7955	. . . . . 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 2732	. . . . 5 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
14	4, 7, 8, 12, 13	fuco1 49353	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
15	1, 14	eqtr3d 2768	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
16	15	oveqd 7358	. 2 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))𝐹))
17		ovres 7507	. . 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 2766	1 ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   × cxp 5609   ↾ cres 5613  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  Catccat 17565   Func cfunc 17756   ∘_func ccofu 17758   ∘_F cfuco 49348

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-func 17760  df-cofu 17762  df-fuco 49349

This theorem is referenced by:  postcofval  49396  precofval  49399