Theorem clsk1indlem4 44847

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))

Assertion

Ref	Expression
clsk1indlem4	⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Distinct variable group:   𝑠,𝑟

Allowed substitution hints:   𝐾(𝑠, 𝑟)

Proof of Theorem clsk1indlem4

Step	Hyp	Ref	Expression
1		tpex 7753	. . . . . . . . . 10 ⊢ {∅, 1o, 2o} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
3		snsstp1 4784	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1o, 2o}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
5		0ex 5272	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4752	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
7	4, 6	sylibr 237	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
8		snsstp2 4785	. . . . . . . . . . . 12 ⊢ {1o} ⊆ {∅, 1o, 2o}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1o} ⊆ {∅, 1o, 2o})
10		1oex 8469	. . . . . . . . . . . 12 ⊢ 1o ∈ V
11	10	snss 4752	. . . . . . . . . . 11 ⊢ (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
12	9, 11	sylibr 237	. . . . . . . . . 10 ⊢ (⊤ → 1o ∈ {∅, 1o, 2o})
13	7, 12	prssd 4790	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
14	2, 13	sselpwd 5301	. . . . . . . 8 ⊢ (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
15	14	mptru 1577	. . . . . . 7 ⊢ {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16		df3o2 44117	. . . . . . . 8 ⊢ 3o = {∅, 1o, 2o}
17	16	pweqi 4580	. . . . . . 7 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
18	15, 17	eleqtrri 2864	. . . . . 6 ⊢ {∅, 1o} ∈ 𝒫 3o
19	18	a1i 11	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → {∅, 1o} ∈ 𝒫 3o)
20		id 23	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ∈ 𝒫 3o)
21	19, 20	ifcld 4536	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22		eqeq1 2769	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23		eqcom 2772	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24		eqif 4531	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
25	23, 24	bitri 278	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	22, 25	bitrdi 290	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))))
27		id 23	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → 𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
28	26, 27	ifbieq2d 4516	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29		1n0 8478	. . . . . . . . . 10 ⊢ 1o ≠ ∅
30		dfsn2 4604	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
31	30	eqeq1i 2770	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
32	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
33		1on 8472	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1o ∈ On)
35	32, 34	preq2b 4814	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
36	35	mptru 1577	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37		eqcom 2772	. . . . . . . . . . 11 ⊢ (∅ = 1o ↔ 1o = ∅)
38	31, 36, 37	3bitri 300	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1o} ↔ 1o = ∅)
39	29, 38	nemtbir 3056	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1o}
40	39	intnan 492	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41		pm3.24 408	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2772	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 637	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 325	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 894	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4499	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
47	28, 46	eqtrdi 2816	. . . . 5 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
48		clsk1indlem.k	. . . . 5 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
49		prex 5411	. . . . . 6 ⊢ {∅, 1o} ∈ V
50		vex 3461	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4540	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 6993	. . . 4 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o → (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
53	21, 52	syl 18	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
54		eqeq1 2769	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 23	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4516	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
57	56, 48, 51	fvmpt 6993	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
58	57	fveq2d 6889	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
59	53, 58, 57	3eqtr4d 2810	. 2 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 3083	1 ⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ifcif 4489  𝒫 cpw 4564  {csn 4591  {cpr 4593  {ctp 4595   ↦ cmpt 5194  Oncon0 6364  ‘cfv 6540  1_oc1o 8452  2_oc2o 8453  3_oc3o 8454

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fv 6548  df-1o 8459  df-2o 8460  df-3o 8461

This theorem is used by:  clsk1independent  44849