Theorem parhomel 50835

Description: Membership in a hom-set of a parallel-arrow category is equivalent to being its identity value on the diagonal, or a member of 𝑆 at the distinguished source and target. (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
parhomel.j	⊢ 𝐽 ∈ V

Assertion

Ref	Expression
parhomel	⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑈𝐻𝑉) ↔ ((𝑈 = 𝑉 ∧ 𝐹 = 𝐽) ∨ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 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:   𝐹(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomel

Step	Hyp	Ref	Expression
1		parhomval.h	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
2		parhomval.s	. . . 4 ⊢ 𝑆 ∈ V
3	1, 2	parhomval 50834	. . 3 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
4	3	eleq2d 2846	. 2 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑈𝐻𝑉) ↔ 𝐹 ∈ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))))
5		elif 4525	. . 3 ⊢ (𝐹 ∈ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) ↔ ((𝑈 = 𝑉 ∧ 𝐹 ∈ {𝐽}) ∨ (¬ 𝑈 = 𝑉 ∧ 𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))))
6		parhomel.j	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	elsn2 4625	. . . . 5 ⊢ (𝐹 ∈ {𝐽} ↔ 𝐹 = 𝐽)
8	7	anbi2i 635	. . . 4 ⊢ ((𝑈 = 𝑉 ∧ 𝐹 ∈ {𝐽}) ↔ (𝑈 = 𝑉 ∧ 𝐹 = 𝐽))
9		elif0 50013	. . . . 5 ⊢ (𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅) ↔ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆))
10	9	anbi2i 635	. . . 4 ⊢ ((¬ 𝑈 = 𝑉 ∧ 𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) ↔ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆)))
11	8, 10	orbi12i 928	. . 3 ⊢ (((𝑈 = 𝑉 ∧ 𝐹 ∈ {𝐽}) ∨ (¬ 𝑈 = 𝑉 ∧ 𝐹 ∈ if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))) ↔ ((𝑈 = 𝑉 ∧ 𝐹 = 𝐽) ∨ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆))))
12	5, 11	bitri 278	. 2 ⊢ (𝐹 ∈ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)) ↔ ((𝑈 = 𝑉 ∧ 𝐹 = 𝐽) ∨ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆))))
13	4, 12	bitrdi 290	1 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑈𝐻𝑉) ↔ ((𝑈 = 𝑉 ∧ 𝐹 = 𝐽) ∨ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = 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:  parhomcl2  50836  parhomne2  50845  parmapcl2  50861