Theorem parhomssun 50952

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 4503	. . . . 5 ⊢ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ ({𝐽} ∪ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅))
2		ssun2 4180	. . . . . 6 ⊢ {𝐽} ⊆ (𝑆 ∪ {𝐽})
3		ifssun 4503	. . . . . . 7 ⊢ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) ⊆ (𝑆 ∪ ∅)
4		un0 4347	. . . . . . . 8 ⊢ (𝑆 ∪ ∅) = 𝑆
5		ssun1 4179	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝐽})
6	4, 5	eqsstri 3959	. . . . . . 7 ⊢ (𝑆 ∪ ∅) ⊆ (𝑆 ∪ {𝐽})
7	3, 6	sstri 3922	. . . . . 6 ⊢ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) ⊆ (𝑆 ∪ {𝐽})
8	2, 7	unssi 4192	. . . . 5 ⊢ ({𝐽} ∪ if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽})
9	1, 8	sstri 3922	. . . 4 ⊢ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) ⊆ (𝑆 ∪ {𝐽})
10	9	rgen2w 3085	. . 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 8129	. . 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 3962	1 ⊢ (𝑈𝐻𝑉) ⊆ 𝐾

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∀wral 3080   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  ifcif 4485  {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-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:  parhomcl  50953