Theorem fuco112 49816

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⟩⟩)
fuco11a.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
fuco112	⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑃(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem fuco112

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		fuco11a.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco11a 49815	. . 3 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
7	6	fveq2d 6838	. 2 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (2nd ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩))
8		relfunc 17820	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9	8	brrelex1i 5680	. . . . 5 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
10	3, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
11		relfunc 17820	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
12	11	brrelex1i 5680	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
13	2, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
14	10, 13	coexd 7875	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) ∈ V)
15	5	fvexi 6848	. . . 4 ⊢ 𝐵 ∈ V
16	15, 15	mpoex 8025	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V
17		op2ndg 7948	. . 3 ⊢ (((𝐾 ∘ 𝐹) ∈ V ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V) → (2nd ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
18	14, 16, 17	sylancl 587	. 2 ⊢ (𝜑 → (2nd ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
19	7, 18	eqtrd 2772	1 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  2^nd c2nd 7934  Basecbs 17170   Func cfunc 17812   ∘_F cfuco 49803

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-ixp 8839  df-func 17816  df-cofu 17818  df-fuco 49804

This theorem is referenced by:  fuco112x  49819