Theorem parhomideq 50955

Description: An endomorphism in the parallel-arrow construction has the identity value. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 16-Sep-2026.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomideq.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parhomideq.e	⊢ (𝜑 → 𝑈 = 𝑉)

Assertion

Ref	Expression
parhomideq	⊢ (𝜑 → 𝐹 = 𝐽)

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:   𝜑(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomideq

Step	Hyp	Ref	Expression
1		parhomideq.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
2		parhomideq.e	. . . . 5 ⊢ (𝜑 → 𝑈 = 𝑉)
3	2	oveq2d 7459	. . . 4 ⊢ (𝜑 → (𝑈𝐻𝑈) = (𝑈𝐻𝑉))
4	1, 3	eleqtrrd 2867	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑈))
5		parhomval.h	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
6	5	elmpocl1 7687	. . . 4 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → 𝑈 ∈ 𝐵)
7	5	parhomidval 50951	. . . 4 ⊢ (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})
8	1, 6, 7	3syl 19	. . 3 ⊢ (𝜑 → (𝑈𝐻𝑈) = {𝐽})
9	4, 8	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝐹 ∈ {𝐽})
10	9	elsnd 4602	1 ⊢ (𝜑 → 𝐹 = 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-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-ne 2960  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:  parhomco2  50965  parhomco  50966  paridval  50970  parcoval2  50971  parmapcl  50976  parmapco2  50977  parmapco  50978