Theorem parhomssun 50838

Description: Every value of the parallel-arrow hom operation is a subclass of the arrow carrier. (Contributed by Zhi Wang, 16-Sep-2026.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomssun.k	⊢ 𝐾 = (𝑆 ∪ {𝐽})

Assertion

Ref	Expression
parhomssun	⊢ (𝑈𝐻𝑉) ⊆ 𝐾

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:   𝐻(𝑥, 𝑦)   𝐾(𝑥, 𝑦)

Proof of Theorem parhomssun

Step	Hyp	Ref	Expression
1		ifssun 4499	. . . . 5 ⊢ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ ({𝐽} ∪ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
2		ssun2 4177	. . . . . 6 ⊢ {𝐽} ⊆ (𝑆 ∪ {𝐽})
3		ifssun 4499	. . . . . . 7 ⊢ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) ⊆ (𝑆 ∪ ∅)
4		un0 4343	. . . . . . . 8 ⊢ (𝑆 ∪ ∅) = 𝑆
5		ssun1 4176	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝐽})
6	4, 5	eqsstri 3954	. . . . . . 7 ⊢ (𝑆 ∪ ∅) ⊆ (𝑆 ∪ {𝐽})
7	3, 6	sstri 3917	. . . . . 6 ⊢ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) ⊆ (𝑆 ∪ {𝐽})
8	2, 7	unssi 4189	. . . . 5 ⊢ ({𝐽} ∪ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽})
9	1, 8	sstri 3917	. . . 4 ⊢ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽})
10	9	rgen2w 3081	. . 3 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽})
11		parhomval.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
12	11	ovmptss 8104	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽}) → (𝑈𝐻𝑉) ⊆ (𝑆 ∪ {𝐽}))
13	10, 12	ax-mp 5	. 2 ⊢ (𝑈𝐻𝑉) ⊆ (𝑆 ∪ {𝐽})
14		parhomssun.k	. 2 ⊢ 𝐾 = (𝑆 ∪ {𝐽})
15	13, 14	sseqtrri 3957	1 ⊢ (𝑈𝐻𝑉) ⊆ 𝐾

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∀wral 3076   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  ifcif 4481  {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-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:  parhomcl  50839