Theorem par2hom 50870

Description: The four hom-sets of the explicit two-object parallel-arrow category. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypothesis

Ref	Expression
par2hom.h	⊢ 𝐻 = (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))

Assertion

Ref	Expression
par2hom	⊢ ((∅𝐻∅) = {2o} ∧ (1o𝐻1o) = {2o} ∧ ((∅𝐻1o) = 2o ∧ (1o𝐻∅) = ∅))

Distinct variable group:   𝑚,𝑛

Allowed substitution hints:   𝐻(𝑚, 𝑛)

Proof of Theorem par2hom

Step	Hyp	Ref	Expression
1		0ex 5260	. . . . . 6 ⊢ ∅ ∈ V
2	1	prid1 4722	. . . . 5 ⊢ ∅ ∈ {∅, 1o}
3		df2o3 8486	. . . . 5 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2859	. . . 4 ⊢ ∅ ∈ 2o
5		par2hom.h	. . . . 5 ⊢ 𝐻 = (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))
6		2oex 8490	. . . . 5 ⊢ 2o ∈ V
7	5, 6	parhomval 50834	. . . 4 ⊢ ((∅ ∈ 2o ∧ ∅ ∈ 2o) → (∅𝐻∅) = if(∅ = ∅, {2o}, if((∅ = ∅ ∧ ∅ = 1o), 2o, ∅)))
8	4, 4, 7	mp2an 705	. . 3 ⊢ (∅𝐻∅) = if(∅ = ∅, {2o}, if((∅ = ∅ ∧ ∅ = 1o), 2o, ∅))
9		eqid 2760	. . . 4 ⊢ ∅ = ∅
10	9	iftruei 4488	. . 3 ⊢ if(∅ = ∅, {2o}, if((∅ = ∅ ∧ ∅ = 1o), 2o, ∅)) = {2o}
11	8, 10	eqtri 2783	. 2 ⊢ (∅𝐻∅) = {2o}
12		1oelpr 8489	. . . . 5 ⊢ 1o ∈ {∅, 1o}
13	12, 3	eleqtrri 2859	. . . 4 ⊢ 1o ∈ 2o
14	5, 6	parhomval 50834	. . . 4 ⊢ ((1o ∈ 2o ∧ 1o ∈ 2o) → (1o𝐻1o) = if(1o = 1o, {2o}, if((1o = ∅ ∧ 1o = 1o), 2o, ∅)))
15	13, 13, 14	mp2an 705	. . 3 ⊢ (1o𝐻1o) = if(1o = 1o, {2o}, if((1o = ∅ ∧ 1o = 1o), 2o, ∅))
16		eqid 2760	. . . 4 ⊢ 1o = 1o
17	16	iftruei 4488	. . 3 ⊢ if(1o = 1o, {2o}, if((1o = ∅ ∧ 1o = 1o), 2o, ∅)) = {2o}
18	15, 17	eqtri 2783	. 2 ⊢ (1o𝐻1o) = {2o}
19	5, 6	parhomval 50834	. . . . 5 ⊢ ((∅ ∈ 2o ∧ 1o ∈ 2o) → (∅𝐻1o) = if(∅ = 1o, {2o}, if((∅ = ∅ ∧ 1o = 1o), 2o, ∅)))
20	4, 13, 19	mp2an 705	. . . 4 ⊢ (∅𝐻1o) = if(∅ = 1o, {2o}, if((∅ = ∅ ∧ 1o = 1o), 2o, ∅))
21		1n0 8497	. . . . . 6 ⊢ 1o ≠ ∅
22	21	nesymi 3012	. . . . 5 ⊢ ¬ ∅ = 1o
23	22	iffalsei 4491	. . . 4 ⊢ if(∅ = 1o, {2o}, if((∅ = ∅ ∧ 1o = 1o), 2o, ∅)) = if((∅ = ∅ ∧ 1o = 1o), 2o, ∅)
24	9, 16	pm3.2i 476	. . . . 5 ⊢ (∅ = ∅ ∧ 1o = 1o)
25	24	iftruei 4488	. . . 4 ⊢ if((∅ = ∅ ∧ 1o = 1o), 2o, ∅) = 2o
26	20, 23, 25	3eqtri 2787	. . 3 ⊢ (∅𝐻1o) = 2o
27	5, 6	parhomval 50834	. . . . 5 ⊢ ((1o ∈ 2o ∧ ∅ ∈ 2o) → (1o𝐻∅) = if(1o = ∅, {2o}, if((1o = ∅ ∧ ∅ = 1o), 2o, ∅)))
28	13, 4, 27	mp2an 705	. . . 4 ⊢ (1o𝐻∅) = if(1o = ∅, {2o}, if((1o = ∅ ∧ ∅ = 1o), 2o, ∅))
29	21	neii 2957	. . . . 5 ⊢ ¬ 1o = ∅
30	29	iffalsei 4491	. . . 4 ⊢ if(1o = ∅, {2o}, if((1o = ∅ ∧ ∅ = 1o), 2o, ∅)) = if((1o = ∅ ∧ ∅ = 1o), 2o, ∅)
31	29	intnanr 493	. . . . 5 ⊢ ¬ (1o = ∅ ∧ ∅ = 1o)
32	31	iffalsei 4491	. . . 4 ⊢ if((1o = ∅ ∧ ∅ = 1o), 2o, ∅) = ∅
33	28, 30, 32	3eqtri 2787	. . 3 ⊢ (1o𝐻∅) = ∅
34	26, 33	pm3.2i 476	. 2 ⊢ ((∅𝐻1o) = 2o ∧ (1o𝐻∅) = ∅)
35	11, 18, 34	3pm3.2i 1358	1 ⊢ ((∅𝐻∅) = {2o} ∧ (1o𝐻1o) = {2o} ∧ ((∅𝐻1o) = 2o ∧ (1o𝐻∅) = ∅))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ifcif 4481  {csn 4583  {cpr 4585  (class class class)co 7420   ∈ cmpo 7422  1_oc1o 8471  2_oc2o 8472

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-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8478  df-2o 8479

This theorem is used by:  par2diag  50871  fuconuniqlem  50872  fuconuniqvalslem  50874