Theorem fuco111 50382

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 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 2, 3, 4, 5	fuco11a 50380	. . 3 ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
7	6	fveq2d 6881	. 2 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (1st ‘⟨(𝐾 ∘ 𝐹), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩))
8		relfunc 18017	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
9	8	brrelex1i 5707	. . . . 5 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
10	3, 9	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
11		relfunc 18017	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
12	11	brrelex1i 5707	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
13	2, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
14	10, 13	coexd 7932	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) ∈ V)
15		fvex 6890	. . . 4 ⊢ (Base‘𝐶) ∈ V
16	15, 15	mpoex 8081	. . 3 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) ∈ V
17		op1stg 8002	. . 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 2796	1 ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   ∘ ccom 5655  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7988  Basecbs 17367   Func cfunc 18009   ∘_F cfuco 50368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-map 8833  df-ixp 8910  df-func 18013  df-cofu 18015  df-fuco 50369

This theorem is used by:  fuco111x  50383  fuco11id  50386  fucocolem4  50408