Theorem fuco112x 49807

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 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco112 49804	. 2 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
7		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
8	7	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑥) = (𝐹‘𝑋))
9		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
10	9	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑌))
11	8, 10	oveq12d 7385	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝐹‘𝑥)𝐿(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
12	7, 9	oveq12d 7385	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
13	11, 12	coeq12d 5819	. 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 7402	. . 3 ⊢ (𝜑 → ((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∈ V)
17		ovexd 7402	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ V)
18	16, 17	coexd 7882	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)) ∈ V)
19	6, 13, 14, 15, 18	ovmpod 7519	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝑂‘𝑈))𝑌) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   class class class wbr 5085   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367  2^nd c2nd 7941  Basecbs 17179   Func cfunc 17821   ∘_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-map 8775  df-ixp 8846  df-func 17825  df-cofu 17827  df-fuco 49792

This theorem is referenced by:  fuco112xa  49808