Theorem parhomideq 50841

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 7436	. . . 4 ⊢ (𝜑 → (𝑈𝐻𝑈) = (𝑈𝐻𝑉))
4	1, 3	eleqtrrd 2863	. . 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 7663	. . . 4 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → 𝑈 ∈ 𝐵)
7	5	parhomidval 50837	. . . 4 ⊢ (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})
8	1, 6, 7	3syl 19	. . 3 ⊢ (𝜑 → (𝑈𝐻𝑈) = {𝐽})
9	4, 8	eleqtrd 2862	. 2 ⊢ (𝜑 → 𝐹 ∈ {𝐽})
10	9	elsnd 4601	1 ⊢ (𝜑 → 𝐹 = 𝐽)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅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-ne 2956  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:  parhomco2  50851  parhomco  50852  paridval  50856  parcoval2  50857  parmapcl  50862  parmapco2  50863  parmapco  50864