Theorem fuco111 50545

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 2760	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco11a 50543	. . 3 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
7	6	fveq2d 6889	. 2 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩))
8		relfunc 18042	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9	8	brrelex1i 5706	. . . . 5 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
10	3, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
11		relfunc 18042	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
12	11	brrelex1i 5706	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
13	2, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
14	10, 13	coexd 7943	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) ∈ V)
15		fvex 6898	. . . 4 ⊢ (Base‘𝐶) ∈ V
16	15, 15	mpoex 8092	. . 3 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V
17		op1stg 8013	. . 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 598	. 2 ⊢ (𝜑 → (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩) = (𝐾 ∘ 𝐹))
19	7, 18	eqtrd 2795	1 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   class class class wbr 5102   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  Basecbs 17392   Func cfunc 18034   ∘_F cfuco 50531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8852  df-ixp 8929  df-func 18038  df-cofu 18040  df-fuco 50532

This theorem is used by:  fuco111x  50546  fuco11id  50549  fucocolem4  50571