Theorem parhomval 50948

Description: At objects in the base, the hom operation has value {𝐽} on the diagonal, 𝑆 from the distinguished source to target, and the empty set otherwise. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

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

Assertion

Ref	Expression
parhomval	⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))

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)

Proof of Theorem parhomval

Step	Hyp	Ref	Expression
1		eqeq12 2780	. . 3 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → (𝑥 = 𝑦 ↔ 𝑈 = 𝑉))
2		simpl 488	. . . . . 6 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → 𝑥 = 𝑈)
3	2	eqeq1d 2765	. . . . 5 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → (𝑥 = 𝑋 ↔ 𝑈 = 𝑋))
4		simpr 490	. . . . . 6 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → 𝑦 = 𝑉)
5	4	eqeq1d 2765	. . . . 5 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → (𝑦 = 𝑌 ↔ 𝑉 = 𝑌))
6	3, 5	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) ↔ (𝑈 = 𝑋 ∧ 𝑉 = 𝑌)))
7	6	ifbid 4509	. . 3 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
8	1, 7	ifbieq2d 4512	. 2 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
9		parhomval.h	. 2 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
10		snex 5401	. . 3 ⊢ {𝐽} ∈ V
11		parhomval.s	. . . 4 ⊢ 𝑆 ∈ V
12		0ex 5264	. . . 4 ⊢ ∅ ∈ V
13	11, 12	ifex 4536	. . 3 ⊢ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅) ∈ V
14	10, 13	ifex 4536	. 2 ⊢ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) ∈ V
15	8, 9, 14	ovmpoa 7599	1 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅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-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  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-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  parhomel  50949  parhomideq2  50954  parhomid2  50956  par2hom  50984