Theorem parhomval 50834

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 2777	. . 3 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → (𝑥 = 𝑦 ↔ 𝑈 = 𝑉))
2		simpl 488	. . . . . 6 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → 𝑥 = 𝑈)
3	2	eqeq1d 2762	. . . . 5 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → (𝑥 = 𝑋 ↔ 𝑈 = 𝑋))
4		simpr 490	. . . . . 6 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → 𝑦 = 𝑉)
5	4	eqeq1d 2762	. . . . 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 4505	. . 3 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑉) → if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅) = if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))
8	1, 7	ifbieq2d 4508	. 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 5396	. . 3 ⊢ {𝐽} ∈ V
11		parhomval.s	. . . 4 ⊢ 𝑆 ∈ V
12		0ex 5260	. . . 4 ⊢ ∅ ∈ V
13	11, 12	ifex 4532	. . 3 ⊢ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅) ∈ V
14	10, 13	ifex 4532	. 2 ⊢ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) ∈ V
15	8, 9, 14	ovmpoa 7575	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 1569   ∈ wcel 2144  Vcvv 3450  ∅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-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  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-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parhomel  50835  parhomideq2  50840  parhomid2  50842  par2hom  50870