Theorem clsk1indlem4 45190

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 7772	. . . . . . . . . 10 ⊢ {∅, 1o, 2o} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
3		snsstp1 4775	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1o, 2o}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
5		0ex 5264	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4745	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
7	4, 6	sylibr 237	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
8		snsstp2 4776	. . . . . . . . . . . 12 ⊢ {1o} ⊆ {∅, 1o, 2o}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1o} ⊆ {∅, 1o, 2o})
10		1oex 8500	. . . . . . . . . . . 12 ⊢ 1o ∈ V
11	10	snss 4745	. . . . . . . . . . 11 ⊢ (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
12	9, 11	sylibr 237	. . . . . . . . . 10 ⊢ (⊤ → 1o ∈ {∅, 1o, 2o})
13	7, 12	prssd 4781	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
14	2, 13	sselpwd 5294	. . . . . . . 8 ⊢ (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
15	14	mptru 1576	. . . . . . 7 ⊢ {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16		df3o2 44461	. . . . . . . 8 ⊢ 3o = {∅, 1o, 2o}
17	16	pweqi 4576	. . . . . . 7 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
18	15, 17	eleqtrri 2863	. . . . . 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 4532	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22		eqeq1 2767	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23		eqcom 2770	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24		eqif 4527	. . . . . . . . 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 4512	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29		1n0 8509	. . . . . . . . . 10 ⊢ 1o ≠ ∅
30		dfsn2 4597	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
31	30	eqeq1i 2768	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
32	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
33		1on 8503	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1o ∈ On)
35	32, 34	preq2b 4805	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
36	35	mptru 1576	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37		eqcom 2770	. . . . . . . . . . 11 ⊢ (∅ = 1o ↔ 1o = ∅)
38	31, 36, 37	3bitri 300	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1o} ↔ 1o = ∅)
39	29, 38	nemtbir 3055	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1o}
40	39	intnan 492	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41		pm3.24 408	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2770	. . . . . . . . . 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 4495	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
47	28, 46	eqtrdi 2814	. . . . 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 5400	. . . . . 6 ⊢ {∅, 1o} ∈ V
50		vex 3458	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4536	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 7004	. . . 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 2767	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 23	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4512	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
57	56, 48, 51	fvmpt 7004	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
58	57	fveq2d 6900	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
59	53, 58, 57	3eqtr4d 2808	. 2 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 3082	1 ⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ⊤wtru 1570   ∈ wcel 2146  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  {csn 4588  {cpr 4590  {ctp 4592   ↦ cmpt 5189  Oncon0 6369  ‘cfv 6547  1_oc1o 8483  2_oc2o 8484  3_oc3o 8485

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

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 2101  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-tp 4593  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 6372  df-on 6373  df-suc 6375  df-iota 6501  df-fun 6549  df-fv 6555  df-1o 8490  df-2o 8491  df-3o 8492

This theorem is used by:  clsk1independent  45192