Theorem fuco112x 49227

Description: The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the morphism part of the composed functor. (Contributed by Zhi Wang, 3-Oct-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⟩⟩)
fuco111x.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco112x.y	⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))

Assertion

Ref	Expression
fuco112x	⊢ (𝜑 → (𝑋(2nd ‘(𝑂‘𝑈))𝑌) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))

Proof of Theorem fuco112x

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuco11.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fuco11.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		fuco11.k	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
4		fuco11.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
5		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco112 49224	. 2 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
7		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
8	7	fveq2d 6869	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋))
9		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
10	9	fveq2d 6869	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌))
11	8, 10	oveq12d 7412	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐿(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
12	7, 9	oveq12d 7412	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
13	11, 12	coeq12d 5836	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))
14		fuco111x.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
15		fuco112x.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
16		ovexd 7429	. . 3 ⊢ (𝜑 → ((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∈ V)
17		ovexd 7429	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ V)
18	16, 17	coexd 7916	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)) ∈ V)
19	6, 13, 14, 15, 18	ovmpod 7548	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝑂‘𝑈))𝑌) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3455  ⟨cop 4603   class class class wbr 5115   ∘ ccom 5650  ‘cfv 6519  (class class class)co 7394  2^nd c2nd 7976  Basecbs 17185   Func cfunc 17822   ∘_F cfuco 49211

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 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-func 17826  df-cofu 17828  df-fuco 49212

This theorem is referenced by:  fuco112xa  49228