Theorem parhomss 50848

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 4803	. . . 4 ⊢ {𝐽} ⊆ 𝒫 𝐽
3	1, 2	unssi 4189	. . 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 2760	. . . 4 ⊢ (𝑆 ∪ {𝐽}) = (𝑆 ∪ {𝐽})
6		parhomss.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
7	4, 5, 6	parhomcl 50839	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑆 ∪ {𝐽}))
8	3, 7	sselid 3906	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐽)
9	8	elpwid 4565	1 ⊢ (𝜑 → 𝐹 ⊆ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ∅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-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-cnv 5658  df-dm 5660  df-rn 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parhomco2  50851  parhomco  50852  parcatid2  50854  parcatid  50855  parmapco2  50863  parmapco  50864