Theorem parhomidval 50951

Description: On the diagonal, the parallel-arrow hom operation has value {𝐽}. (Contributed by Zhi Wang, 16-Sep-2026.)

Hypothesis

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

Assertion

Ref	Expression
parhomidval	⊢ (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐻(𝑥, 𝑦)

Proof of Theorem parhomidval

Step	Hyp	Ref	Expression
1		eqtr3 2785	. . . 4 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑈) → 𝑥 = 𝑦)
2	1	iftrued 4493	. . 3 ⊢ ((𝑥 = 𝑈 ∧ 𝑦 = 𝑈) → if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)) = {𝐽})
3		parhomval.h	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
4		snex 5401	. . 3 ⊢ {𝐽} ∈ V
5	2, 3, 4	ovmpoa 7599	. 2 ⊢ ((𝑈 ∈ 𝐵 ∧ 𝑈 ∈ 𝐵) → (𝑈𝐻𝑈) = {𝐽})
6	5	anidms 577	1 ⊢ (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∅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-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:  parhomideq  50955  parhomid  50957