Theorem parswapprop 50868

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 8490	. . . . . 6 ⊢ 2o ∈ V
2	1	sucid 6447	. . . . 5 ⊢ 2o ∈ suc 2o
3		df-3o 8480	. . . . 5 ⊢ 3o = suc 2o
4	2, 3	eleqtrri 2859	. . . 4 ⊢ 2o ∈ 3o
5		parswapval.a	. . . . 5 ⊢ 𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))
6	5	parswapval 50867	. . . 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 2760	. . . 4 ⊢ 2o = 2o
9	8	iftruei 4488	. . 3 ⊢ if(2o = 2o, 2o, if(2o = ∅, 1o, ∅)) = 2o
10	7, 9	eqtri 2783	. 2 ⊢ (𝐴‘2o) = 2o
11		elelsuc 6438	. . . . . . 7 ⊢ (𝑡 ∈ 2o → 𝑡 ∈ suc 2o)
12	11, 3	eleqtrrdi 2871	. . . . . 6 ⊢ (𝑡 ∈ 2o → 𝑡 ∈ 3o)
13	5	parswapval 50867	. . . . . 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 8492	. . . . . . . 8 ⊢ 2o ∈ On
16	15	onirri 6477	. . . . . . 7 ⊢ ¬ 2o ∈ 2o
17		nelneq 2884	. . . . . . 7 ⊢ ((𝑡 ∈ 2o ∧ ¬ 2o ∈ 2o) → ¬ 𝑡 = 2o)
18	16, 17	mpan2 704	. . . . . 6 ⊢ (𝑡 ∈ 2o → ¬ 𝑡 = 2o)
19	18	iffalsed 4492	. . . . 5 ⊢ (𝑡 ∈ 2o → if(𝑡 = 2o, 2o, if(𝑡 = ∅, 1o, ∅)) = if(𝑡 = ∅, 1o, ∅))
20	14, 19	eqtrd 2795	. . . 4 ⊢ (𝑡 ∈ 2o → (𝐴‘𝑡) = if(𝑡 = ∅, 1o, ∅))
21		1oelpr 8489	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
22		df2o3 8486	. . . . . 6 ⊢ 2o = {∅, 1o}
23	21, 22	eleqtrri 2859	. . . . 5 ⊢ 1o ∈ 2o
24		0ex 5260	. . . . . . 7 ⊢ ∅ ∈ V
25	24	prid1 4722	. . . . . 6 ⊢ ∅ ∈ {∅, 1o}
26	25, 22	eleqtrri 2859	. . . . 5 ⊢ ∅ ∈ 2o
27	23, 26	ifcli 4529	. . . 4 ⊢ if(𝑡 = ∅, 1o, ∅) ∈ 2o
28	20, 27	eqeltrdi 2868	. . 3 ⊢ (𝑡 ∈ 2o → (𝐴‘𝑡) ∈ 2o)
29	28	rgen 3078	. 2 ⊢ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o
30		elelsuc 6438	. . . . . 6 ⊢ (∅ ∈ 2o → ∅ ∈ suc 2o)
31	26, 30	ax-mp 5	. . . . 5 ⊢ ∅ ∈ suc 2o
32	31, 3	eleqtrri 2859	. . . 4 ⊢ ∅ ∈ 3o
33	5	parswapval 50867	. . . 4 ⊢ (∅ ∈ 3o → (𝐴‘∅) = if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅)))
34	32, 33	ax-mp 5	. . 3 ⊢ (𝐴‘∅) = if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅))
35		nelneq 2884	. . . . 5 ⊢ ((∅ ∈ 2o ∧ ¬ 2o ∈ 2o) → ¬ ∅ = 2o)
36	26, 16, 35	mp2an 705	. . . 4 ⊢ ¬ ∅ = 2o
37	36	iffalsei 4491	. . 3 ⊢ if(∅ = 2o, 2o, if(∅ = ∅, 1o, ∅)) = if(∅ = ∅, 1o, ∅)
38		eqid 2760	. . . 4 ⊢ ∅ = ∅
39	38	iftruei 4488	. . 3 ⊢ if(∅ = ∅, 1o, ∅) = 1o
40	34, 37, 39	3eqtri 2787	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∅c0 4278  ifcif 4481  {cpr 4585   ↦ cmpt 5185  suc csuc 6364  ‘cfv 6538  1_oc1o 8471  2_oc2o 8472  3_oc3o 8473

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-3or 1104  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-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fv 6546  df-1o 8478  df-2o 8479  df-3o 8480

This theorem is used by:  parswapfunc  50869  fuconuniqlem  50872  fuconuniqvalslem  50874