Theorem fuco11 49801

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 49554	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fuco11.k	. . . . 5 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
4	3	funcrcl2 49554	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	3	funcrcl3 49555	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		fuco11.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
7		eqidd 2737	. . . 4 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8	2, 4, 5, 6, 7	fuco1 49796	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
9	8	fveq1d 6842	. 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 49788	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12	11	fvresd 6860	. 2 ⊢ (𝜑 → (( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))‘𝑈) = ( ∘func ‘𝑈))
13	10	fveq2d 6844	. . 3 ⊢ (𝜑 → ( ∘func ‘𝑈) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
14		df-ov 7370	. . 3 ⊢ (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
15	13, 14	eqtr4di 2789	. 2 ⊢ (𝜑 → ( ∘func ‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))
16	9, 12, 15	3eqtrd 2775	1 ⊢ (𝜑 → (𝑂‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))

Syntax hints:   → wi 4   = wceq 1542  ⟨cop 4573   class class class wbr 5085   × cxp 5629   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  Catccat 17630   Func cfunc 17821   ∘_func ccofu 17823   ∘_F cfuco 49791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-func 17825  df-cofu 17827  df-fuco 49792

This theorem is referenced by:  fuco11a  49803  fuco11bALT  49813  precofvalALT  49843