Theorem parhomco 50852

Description: When every member of the parallel-arrow class 𝑆 is a subset of 𝐽, intersection of composable arrows belongs to the hom-set from the first object to the last. The subset hypothesis supplies the required intersection identity laws; it does not follow from the hom construction alone. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomeqor.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parhomeqor.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
parhomco.p	⊢ 𝑆 ⊆ 𝒫 𝐽

Assertion

Ref	Expression
parhomco	⊢ (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomco

Step	Hyp	Ref	Expression
1		parhomeqor.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 ∈ (𝑉𝐻𝑊))
3		parhomval.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
4		parhomeqor.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
6		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑈 = 𝑉)
7	3, 5, 6	parhomideq 50841	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = 𝐽)
8	7	ineq2d 4117	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = (𝐺 ∩ 𝐽))
9		parhomco.p	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 𝐽
10	3, 9, 1	parhomss 50848	. . . . . 6 ⊢ (𝜑 → 𝐺 ⊆ 𝐽)
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 ⊆ 𝐽)
12		dfss2 4094	. . . . 5 ⊢ (𝐺 ⊆ 𝐽 ↔ (𝐺 ∩ 𝐽) = 𝐺)
13	11, 12	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐽) = 𝐺)
14	8, 13	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) = 𝐺)
15	6	oveq1d 7435	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑈𝐻𝑊) = (𝑉𝐻𝑊))
16	2, 14, 15	3eltr4d 2875	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
17	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 ∈ (𝑉𝐻𝑊))
18		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝑉 = 𝑊)
19	3, 17, 18	parhomideq 50841	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐺 = 𝐽)
20	19	ineq1d 4116	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = (𝐽 ∩ 𝐹))
21	3, 9, 4	parhomss 50848	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝐽)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ⊆ 𝐽)
23		sseqin2 4120	. . . . 5 ⊢ (𝐹 ⊆ 𝐽 ↔ (𝐽 ∩ 𝐹) = 𝐹)
24	22, 23	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐽 ∩ 𝐹) = 𝐹)
25	20, 24	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) = 𝐹)
26	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ∈ (𝑈𝐻𝑉))
27	18	oveq2d 7436	. . . 4 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝑈𝐻𝑉) = (𝑈𝐻𝑊))
28	26, 27	eleqtrd 2862	. . 3 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → 𝐹 ∈ (𝑈𝐻𝑊))
29	25, 28	eqeltrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑉 = 𝑊) → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
30	3, 4, 1	parhomeqor 50850	. 2 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
31	16, 29, 30	mpjaodan 973	1 ⊢ (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ifcif 4481  𝒫 cpw 4556  {csn 4583  (class class class)co 7420   ∈ cmpo 7422

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-sep 5248  ax-nul 5259  ax-pr 5390

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-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parcatid2  50854  parcatid  50855  parmapfunc2  50865