Theorem parhomeqor 50850

Description: In a composable pair, at least one arrow is an endomorphism. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomeqor.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parhomeqor.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))

Assertion

Ref	Expression
parhomeqor	⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomeqor

Step	Hyp	Ref	Expression
1		parhomval.h	. . . . . . . 8 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
2		parhomeqor.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
3	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
4		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ¬ 𝑈 = 𝑉)
5	1, 3, 4	parhomne 50846	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈 = 𝑋 ∧ 𝑉 = 𝑌))
6	5	simpld 500	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑈 = 𝑋)
7	6	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑈 = 𝑋)
8		parhomeqor.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐺 ∈ (𝑉𝐻𝑊))
10	9	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝐺 ∈ (𝑉𝐻𝑊))
11		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → ¬ 𝑉 = 𝑊)
12	1, 10, 11	parhomne 50846	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → (𝑉 = 𝑋 ∧ 𝑊 = 𝑌))
13	12	simpld 500	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑉 = 𝑋)
14	7, 13	eqtr4d 2798	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑈 = 𝑉)
15	4	adantr 486	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → ¬ 𝑈 = 𝑉)
16	14, 15	condan 830	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑉 = 𝑊)
17	16	ex 418	. 2 ⊢ (𝜑 → (¬ 𝑈 = 𝑉 → 𝑉 = 𝑊))
18	17	orrd 877	1 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = 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-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-xp 5656  df-dm 5660  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  parhomco2  50851  parhomco  50852  parcoval2  50857  parmapco2  50863  parmapco  50864