Theorem parhomel 50949

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 50948	. . 3 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
4	3	eleq2d 2850	. 2 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑈𝐻𝑉) ↔ 𝐹 ∈ if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅))))
5		elif 4529	. . 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 4626	. . . . 5 ⊢ (𝐹 ∈ {𝐽} ↔ 𝐹 = 𝐽)
8	7	anbi2i 635	. . . 4 ⊢ ((𝑈 = 𝑉 ∧ 𝐹 ∈ {𝐽}) ↔ (𝑈 = 𝑉 ∧ 𝐹 = 𝐽))
9		elif0 50127	. . . . 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 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:  parhomcl2  50950  parhomne2  50959  parmapcl2  50975