Theorem parcoval 50967

Description: The composition in the specified parallel-arrow categories is intersection. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
parcoval.m	⊢ 𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))
parcoval.c	⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ 𝑀)
parcoval.s	⊢ 𝑀 ∈ V
parcoval.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
parcoval.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
parcoval.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
parcoval.f	⊢ (𝜑 → 𝐹 ∈ 𝐾)
parcoval.g	⊢ (𝜑 → 𝐺 ∈ 𝐾)

Assertion

Ref	Expression
parcoval	⊢ (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (𝐺 ∩ 𝐹))

Distinct variable groups:   𝐵,𝑓,𝑔,𝑝,𝑧   𝑓,𝐾,𝑔,𝑝,𝑧   𝑈,𝑓,𝑔,𝑝,𝑧   𝑓,𝑉,𝑔,𝑝,𝑧   𝑓,𝑊,𝑔,𝑝,𝑧   𝑓,𝐹,𝑔,𝑝,𝑧   𝑓,𝐺,𝑔,𝑝,𝑧   𝜑,𝑓,𝑔,𝑝,𝑧   𝑀,𝑝,𝑧

Allowed substitution hints:   · (𝑧, 𝑓, 𝑔, 𝑝)   𝑀(𝑓, 𝑔)

Proof of Theorem parcoval

Step	Hyp	Ref	Expression
1		parcoval.c	. . . . 5 ⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ 𝑀)
2	1	a1i 11	. . . 4 ⊢ (𝜑 → · = (𝑝 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ 𝑀))
3		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝑈, 𝑉⟩ ∧ 𝑧 = 𝑊)) → 𝑀 = 𝑀)
4		parcoval.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
5		parcoval.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
6	4, 5	opelxpd 5694	. . . 4 ⊢ (𝜑 → ⟨𝑈, 𝑉⟩ ∈ (𝐵 × 𝐵))
7		parcoval.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
8		parcoval.s	. . . . 5 ⊢ 𝑀 ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ∈ V)
10	2, 3, 6, 7, 9	ovmpod 7596	. . 3 ⊢ (𝜑 → (⟨𝑈, 𝑉⟩ · 𝑊) = 𝑀)
11	10	oveqd 7460	. 2 ⊢ (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (𝐺𝑀𝐹))
12		parcoval.m	. . . 4 ⊢ 𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))
13	12	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔)))
14		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑓 = 𝐺)
15		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑔 = 𝐹)
16	14, 15	ineq12d 4121	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐺 ∧ 𝑔 = 𝐹)) → (𝑓 ∩ 𝑔) = (𝐺 ∩ 𝐹))
17		parcoval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐾)
18		parcoval.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐾)
19		inex1g 5283	. . . 4 ⊢ (𝐺 ∈ 𝐾 → (𝐺 ∩ 𝐹) ∈ V)
20	17, 19	syl 18	. . 3 ⊢ (𝜑 → (𝐺 ∩ 𝐹) ∈ V)
21	13, 16, 17, 18, 20	ovmpod 7596	. 2 ⊢ (𝜑 → (𝐺𝑀𝐹) = (𝐺 ∩ 𝐹))
22	11, 21	eqtrd 2798	1 ⊢ (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (𝐺 ∩ 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ⟨cop 4838   × cxp 5653  (class class class)co 7443   ∈ cmpo 7445

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  parcatid2  50968  parcatid  50969  parmapfunc2  50979