Theorem fuco111 49316

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 2729	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco11a 49314	. . 3 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
7	6	fveq2d 6830	. 2 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩))
8		relfunc 17787	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9	8	brrelex1i 5679	. . . . 5 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
10	3, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
11		relfunc 17787	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
12	11	brrelex1i 5679	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
13	2, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
14	10, 13	coexd 7871	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) ∈ V)
15		fvex 6839	. . . 4 ⊢ (Base‘𝐶) ∈ V
16	15, 15	mpoex 8021	. . 3 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V
17		op1stg 7943	. . 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 2764	1 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ⟨cop 4585   class class class wbr 5095   ∘ ccom 5627  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  Basecbs 17138   Func cfunc 17779   ∘_F cfuco 49302

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-ixp 8832  df-func 17783  df-cofu 17785  df-fuco 49303

This theorem is referenced by:  fuco111x  49317  fuco11id  49320  fucocolem4  49342