Theorem fuco112xa 49228

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. A morphism is mapped by two functors in succession. (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))
fuco112xa.a	⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝐶)𝑌))

Assertion

Ref	Expression
fuco112xa	⊢ (𝜑 → ((𝑋(2nd ‘(𝑂‘𝑈))𝑌)‘𝐴) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐴)))

Proof of Theorem fuco112xa

Step	Hyp	Ref	Expression
1		fuco11.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fuco11.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		fuco11.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
4		fuco11.u	. . . 4 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
5		fuco111x.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
6		fuco112x.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
7	1, 2, 3, 4, 5, 6	fuco112x 49227	. . 3 ⊢ (𝜑 → (𝑋(2nd ‘(𝑂‘𝑈))𝑌) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))
8	7	fveq1d 6867	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘(𝑂‘𝑈))𝑌)‘𝐴) = ((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐴))
9		eqid 2730	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
10		eqid 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11		eqid 2730	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
12	9, 10, 11, 2, 5, 6	funcf2 17836	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
13		fuco112xa.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝐶)𝑌))
14	12, 13	fvco3d 6968	. 2 ⊢ (𝜑 → ((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐴) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐴)))
15	8, 14	eqtrd 2765	1 ⊢ (𝜑 → ((𝑋(2nd ‘(𝑂‘𝑈))𝑌)‘𝐴) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4603   class class class wbr 5115   ∘ ccom 5650  ‘cfv 6519  (class class class)co 7394  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237   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: (None)