Theorem clsk1indlem4 44062

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 7767	. . . . . . . . . 10 ⊢ {∅, 1o, 2o} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
3		snsstp1 4815	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1o, 2o}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
5		0ex 5306	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4784	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
7	4, 6	sylibr 234	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
8		snsstp2 4816	. . . . . . . . . . . 12 ⊢ {1o} ⊆ {∅, 1o, 2o}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1o} ⊆ {∅, 1o, 2o})
10		1oex 8517	. . . . . . . . . . . 12 ⊢ 1o ∈ V
11	10	snss 4784	. . . . . . . . . . 11 ⊢ (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
12	9, 11	sylibr 234	. . . . . . . . . 10 ⊢ (⊤ → 1o ∈ {∅, 1o, 2o})
13	7, 12	prssd 4821	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
14	2, 13	sselpwd 5327	. . . . . . . 8 ⊢ (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
15	14	mptru 1546	. . . . . . 7 ⊢ {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16		df3o2 43331	. . . . . . . 8 ⊢ 3o = {∅, 1o, 2o}
17	16	pweqi 4615	. . . . . . 7 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
18	15, 17	eleqtrri 2839	. . . . . 6 ⊢ {∅, 1o} ∈ 𝒫 3o
19	18	a1i 11	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → {∅, 1o} ∈ 𝒫 3o)
20		id 22	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ∈ 𝒫 3o)
21	19, 20	ifcld 4571	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22		eqeq1 2740	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23		eqcom 2743	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24		eqif 4566	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
25	23, 24	bitri 275	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	22, 25	bitrdi 287	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))))
27		id 22	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → 𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
28	26, 27	ifbieq2d 4551	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29		1n0 8527	. . . . . . . . . 10 ⊢ 1o ≠ ∅
30		dfsn2 4638	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
31	30	eqeq1i 2741	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
32	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
33		1on 8519	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1o ∈ On)
35	32, 34	preq2b 4846	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
36	35	mptru 1546	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37		eqcom 2743	. . . . . . . . . . 11 ⊢ (∅ = 1o ↔ 1o = ∅)
38	31, 36, 37	3bitri 297	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1o} ↔ 1o = ∅)
39	29, 38	nemtbir 3037	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1o}
40	39	intnan 486	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41		pm3.24 402	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2743	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 625	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 322	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 880	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4534	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
47	28, 46	eqtrdi 2792	. . . . 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 5436	. . . . . 6 ⊢ {∅, 1o} ∈ V
50		vex 3483	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4575	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 7015	. . . 4 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o → (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
53	21, 52	syl 17	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
54		eqeq1 2740	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 22	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4551	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
57	56, 48, 51	fvmpt 7015	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
58	57	fveq2d 6909	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
59	53, 58, 57	3eqtr4d 2786	. 2 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 3062	1 ⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ⊆ wss 3950  ∅c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625  {cpr 4627  {ctp 4629   ↦ cmpt 5224  Oncon0 6383  ‘cfv 6560  1_oc1o 8500  2_oc2o 8501  3_oc3o 8502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fv 6568  df-1o 8507  df-2o 8508  df-3o 8509

This theorem is referenced by:  clsk1independent  44064