Theorem fuco11 50260

Description: The object part of the functor composition bifunctor maps two functors to their composition. (Contributed by Zhi Wang, 30-Sep-2025.)

Hypotheses

Ref	Expression
fuco11.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco11.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
fuco11.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
fuco11.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)

Assertion

Ref	Expression
fuco11	⊢ (𝜑 → (𝑂‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))

Proof of Theorem fuco11

Step	Hyp	Ref	Expression
1		fuco11.f	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
2	1	funcrcl2 50013	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fuco11.k	. . . . 5 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
4	3	funcrcl2 50013	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	3	funcrcl3 50014	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		fuco11.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
7		eqidd 2763	. . . 4 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8	2, 4, 5, 6, 7	fuco1 50255	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
9	8	fveq1d 6884	. 2 ⊢ (𝜑 → (𝑂‘𝑈) = (( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))‘𝑈))
10		fuco11.u	. . . 4 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
11	7, 10, 3, 1	fuco2eld 50247	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12	11	fvresd 6902	. 2 ⊢ (𝜑 → (( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))‘𝑈) = ( ∘func ‘𝑈))
13	10	fveq2d 6886	. . 3 ⊢ (𝜑 → ( ∘func ‘𝑈) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
14		df-ov 7420	. . 3 ⊢ (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
15	13, 14	eqtr4di 2815	. 2 ⊢ (𝜑 → ( ∘func ‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))
16	9, 12, 15	3eqtrd 2801	1 ⊢ (𝜑 → (𝑂‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨cop 4593   class class class wbr 5107   × cxp 5657   ↾ cres 5661  ‘cfv 6537  (class class class)co 7417  Catccat 17758   Func cfunc 17949   ∘_func ccofu 17951   ∘_F cfuco 50250

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-func 17953  df-cofu 17955  df-fuco 50251

This theorem is used by:  fuco11a  50262  fuco11bALT  50272  precofvalALT  50302