Theorem parhomss 50962

Description: When every member of the parallel-arrow class 𝑆 is a subset of 𝐽, every arrow in a hom-set is a subset of its identity value 𝐽. The subset hypothesis is essential; it does not follow from the hom-set definition 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, ∅)))
parhomss.p	⊢ 𝑆 ⊆ 𝒫 𝐽
parhomss.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))

Assertion

Ref	Expression
parhomss	⊢ (𝜑 → 𝐹 ⊆ 𝐽)

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

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

Proof of Theorem parhomss

Step	Hyp	Ref	Expression
1		parhomss.p	. . . 4 ⊢ 𝑆 ⊆ 𝒫 𝐽
2		snsspw 4802	. . . 4 ⊢ {𝐽} ⊆ 𝒫 𝐽
3	1, 2	unssi 4192	. . 3 ⊢ (𝑆 ∪ {𝐽}) ⊆ 𝒫 𝐽
4		parhomval.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
5		eqid 2763	. . . 4 ⊢ (𝑆 ∪ {𝐽}) = (𝑆 ∪ {𝐽})
6		parhomss.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
7	4, 5, 6	parhomcl 50953	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 ∪ {𝐽}))
8	3, 7	sselid 3911	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐽)
9	8	elpwid 4569	1 ⊢ (𝜑 → 𝐹 ⊆ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  {csn 4588  (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-nul 5263  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-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-cnv 5663  df-dm 5665  df-rn 5666  df-iota 6512  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  parhomco2  50965  parhomco  50966  parcatid2  50968  parcatid  50969  parmapco2  50977  parmapco  50978