Theorem fuco11 49299

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 49052	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fuco11.k	. . . . 5 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
4	3	funcrcl2 49052	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5	3	funcrcl3 49053	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		fuco11.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
7		eqidd 2730	. . . 4 ⊢ (𝜑 → ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
8	2, 4, 5, 6, 7	fuco1 49294	. . 3 ⊢ (𝜑 → 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))))
9	8	fveq1d 6828	. 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 49286	. . 3 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12	11	fvresd 6846	. 2 ⊢ (𝜑 → (( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))‘𝑈) = ( ∘func ‘𝑈))
13	10	fveq2d 6830	. . 3 ⊢ (𝜑 → ( ∘func ‘𝑈) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
14		df-ov 7356	. . 3 ⊢ (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ( ∘func ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
15	13, 14	eqtr4di 2782	. 2 ⊢ (𝜑 → ( ∘func ‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))
16	9, 12, 15	3eqtrd 2768	1 ⊢ (𝜑 → (𝑂‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))

Syntax hints:   → wi 4   = wceq 1540  ⟨cop 4585   class class class wbr 5095   × cxp 5621   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353  Catccat 17588   Func cfunc 17779   ∘_func ccofu 17781   ∘_F cfuco 49289

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-func 17783  df-cofu 17785  df-fuco 49290

This theorem is referenced by:  fuco11a  49301  fuco11bALT  49311  precofvalALT  49341