Theorem parswapprop 50982

Description: The concrete transposition fixes the identity, preserves the parallel arrows, and sends the empty-set arrow to the ordinal-one arrow. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypothesis

Ref	Expression
parswapval.a	⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))

Assertion

Ref	Expression
parswapprop	⊢ ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)

Distinct variable groups:   𝑡,𝑏   𝑡,𝐴

Allowed substitution hint:   𝐴(𝑏)

Proof of Theorem parswapprop

Step	Hyp	Ref	Expression
1		2oex 8516	. . . . . 6 ⊢ 2o ∈ V
2	1	sucid 6465	. . . . 5 ⊢ 2o ∈ suc 2o
3		df-3o 8506	. . . . 5 ⊢ 3o = suc 2o
4	2, 3	eleqtrri 2863	. . . 4 ⊢ 2o ∈ 3o
5		parswapval.a	. . . . 5 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
6	5	parswapval 50981	. . . 4 ⊢ (2o ∈ 3o → (𝐴‘2o) = if(2o = 2o, 2o, if(2o = ∅, 1o, ∅)))
7	4, 6	ax-mp 5	. . 3 ⊢ (𝐴‘2o) = if(2o = 2o, 2o, if(2o = ∅, 1o, ∅))
8		eqid 2763	. . . 4 ⊢ 2o = 2o
9	8	iftruei 4492	. . 3 ⊢ if(2o = 2o, 2o, if(2o = ∅, 1o, ∅)) = 2o
10	7, 9	eqtri 2786	. 2 ⊢ (𝐴‘2o) = 2o
11		elelsuc 6456	. . . . . . 7 ⊢ (𝑡 ∈ 2o → 𝑡 ∈ suc 2o)
12	11, 3	eleqtrrdi 2875	. . . . . 6 ⊢ (𝑡 ∈ 2o → 𝑡 ∈ 3o)
13	5	parswapval 50981	. . . . . 6 ⊢ (𝑡 ∈ 3o → (𝐴‘𝑡) = if(𝑡 = 2o, 2o, if(𝑡 = ∅, 1o, ∅)))
14	12, 13	syl 18	. . . . 5 ⊢ (𝑡 ∈ 2o → (𝐴‘𝑡) = if(𝑡 = 2o, 2o, if(𝑡 = ∅, 1o, ∅)))
15		2on 8518	. . . . . . . 8 ⊢ 2o ∈ On
16	15	onirri 6495	. . . . . . 7 ⊢ ¬ 2o ∈ 2o
17		nelneq 2888	. . . . . . 7 ⊢ ((𝑡 ∈ 2o ∧ ¬ 2o ∈ 2o) → ¬ 𝑡 = 2o)
18	16, 17	mpan2 704	. . . . . 6 ⊢ (𝑡 ∈ 2o → ¬ 𝑡 = 2o)
19	18	iffalsed 4496	. . . . 5 ⊢ (𝑡 ∈ 2o → if(𝑡 = 2o, 2o, if(𝑡 = ∅, 1o, ∅)) = if(𝑡 = ∅, 1o, ∅))
20	14, 19	eqtrd 2798	. . . 4 ⊢ (𝑡 ∈ 2o → (𝐴‘𝑡) = if(𝑡 = ∅, 1o, ∅))
21		1oelpr 8515	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
22		df2o3 8512	. . . . . 6 ⊢ 2o = {∅, 1o}
23	21, 22	eleqtrri 2863	. . . . 5 ⊢ 1o ∈ 2o
24		0ex 5264	. . . . . . 7 ⊢ ∅ ∈ V
25	24	prid1 4723	. . . . . 6 ⊢ ∅ ∈ {∅, 1o}
26	25, 22	eleqtrri 2863	. . . . 5 ⊢ ∅ ∈ 2o
27	23, 26	ifcli 4533	. . . 4 ⊢ if(𝑡 = ∅, 1o, ∅) ∈ 2o
28	20, 27	eqeltrdi 2872	. . 3 ⊢ (𝑡 ∈ 2o → (𝐴‘𝑡) ∈ 2o)
29	28	rgen 3082	. 2 ⊢ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o
30		elelsuc 6456	. . . . . 6 ⊢ (∅ ∈ 2o → ∅ ∈ suc 2o)
31	26, 30	ax-mp 5	. . . . 5 ⊢ ∅ ∈ suc 2o
32	31, 3	eleqtrri 2863	. . . 4 ⊢ ∅ ∈ 3o
33	5	parswapval 50981	. . . 4 ⊢ (∅ ∈ 3o → (𝐴‘∅) = if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅)))
34	32, 33	ax-mp 5	. . 3 ⊢ (𝐴‘∅) = if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅))
35		nelneq 2888	. . . . 5 ⊢ ((∅ ∈ 2o ∧ ¬ 2o ∈ 2o) → ¬ ∅ = 2o)
36	26, 16, 35	mp2an 705	. . . 4 ⊢ ¬ ∅ = 2o
37	36	iffalsei 4495	. . 3 ⊢ if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅)) = if(∅ = ∅, 1o, ∅)
38		eqid 2763	. . . 4 ⊢ ∅ = ∅
39	38	iftruei 4492	. . 3 ⊢ if(∅ = ∅, 1o, ∅) = 1o
40	34, 37, 39	3eqtri 2790	. 2 ⊢ (𝐴‘∅) = 1o
41	10, 29, 40	3pm3.2i 1358	1 ⊢ ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)

This proof depends on syntax axioms:  ¬ wn 3   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∅c0 4281  ifcif 4485  {cpr 4590   ↦ cmpt 5189  suc csuc 6382  ‘cfv 6558  1_oc1o 8497  2_oc2o 8498  3_oc3o 8499

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-3or 1104  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-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fv 6566  df-1o 8504  df-2o 8505  df-3o 8506

This theorem is used by:  parswapfunc  50983  fuconuniqlem  50986  fuconuniqvalslem  50988