Theorem fuco111 48989

Description: The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the object part of the composed functor. (Contributed by Zhi Wang, 2-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⟩⟩)

Assertion

Ref	Expression
fuco111	⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))

Proof of Theorem fuco111

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

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		eqid 2734	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco11a 48987	. . 3 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
7	6	fveq2d 6891	. 2 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩))
8		relfunc 17883	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9	8	brrelex1i 5723	. . . . 5 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
10	3, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
11		relfunc 17883	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
12	11	brrelex1i 5723	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
13	2, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
14	10, 13	coexd 7936	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) ∈ V)
15		fvex 6900	. . . 4 ⊢ (Base‘𝐶) ∈ V
16	15, 15	mpoex 8087	. . 3 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V
17		op1stg 8009	. . 3 ⊢ (((𝐾 ∘ 𝐹) ∈ V ∧ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V) → (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩) = (𝐾 ∘ 𝐹))
18	14, 16, 17	sylancl 586	. 2 ⊢ (𝜑 → (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩) = (𝐾 ∘ 𝐹))
19	7, 18	eqtrd 2769	1 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3464  ⟨cop 4614   class class class wbr 5125   ∘ ccom 5671  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7995  Basecbs 17230   Func cfunc 17875   ∘_F cfuco 48975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-map 8851  df-ixp 8921  df-func 17879  df-cofu 17881  df-fuco 48976

This theorem is referenced by:  fuco111x  48990  fuco11id  48993  fucocolem4  49015