Theorem clsk1indlem4 37362

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 6855	. . . . . . . . . 10 ⊢ {∅, 1𝑜, 2𝑜} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3		snsstp1 4287	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1𝑜, 2𝑜}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5		0ex 4718	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4259	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
7	4, 6	sylibr 223	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8		snsstp2 4288	. . . . . . . . . . . 12 ⊢ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10		1on 7454	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ On
11	10	elexi 3186	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
12	11	snss 4259	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
13	9, 12	sylibr 223	. . . . . . . . . 10 ⊢ (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
14	7, 13	prssd 4294	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15	2, 14	sselpwd 4734	. . . . . . . 8 ⊢ (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
16	15	trud 1484	. . . . . . 7 ⊢ {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
17		df3o2 37342	. . . . . . . 8 ⊢ 3𝑜 = {∅, 1𝑜, 2𝑜}
18	17	pweqi 4112	. . . . . . 7 ⊢ 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
19	16, 18	eleqtrri 2687	. . . . . 6 ⊢ {∅, 1𝑜} ∈ 𝒫 3𝑜
20	19	a1i 11	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3𝑜 → {∅, 1𝑜} ∈ 𝒫 3𝑜)
21		id 22	. . . . 5 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ∈ 𝒫 3𝑜)
22	20, 21	ifcld 4081	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
23		eqeq1 2614	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
24		eqcom 2617	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
25		eqif 4076	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	24, 25	bitri 263	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
27	23, 26	syl6bb 275	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))))
28		id 22	. . . . . . 7 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → 𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
29	27, 28	ifbieq2d 4061	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
30		1n0 7462	. . . . . . . . . 10 ⊢ 1𝑜 ≠ ∅
31		dfsn2 4138	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
32	31	eqeq1i 2615	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
33	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
34	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1𝑜 ∈ On)
35	33, 34	preq2b 4318	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
36	35	trud 1484	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37		eqcom 2617	. . . . . . . . . . 11 ⊢ (∅ = 1𝑜 ↔ 1𝑜 = ∅)
38	32, 36, 37	3bitri 285	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
39	30, 38	nemtbir 2877	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1𝑜}
40	39	intnan 951	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41		pm3.24 922	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2617	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 728	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 311	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 895	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4046	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
47	29, 46	syl6eq 2660	. . . . 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 4836	. . . . . 6 ⊢ {∅, 1𝑜} ∈ V
50		vex 3176	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4106	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 6191	. . . 4 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
53	22, 52	syl 17	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
54		eqeq1 2614	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 22	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4061	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
57	56, 48, 51	fvmpt 6191	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
58	57	fveq2d 6107	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
59	53, 58, 57	3eqtr4d 2654	. 2 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 2906	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127  {ctp 4129   ↦ cmpt 4643  Oncon0 5640  ‘cfv 5804  1_𝑜c1o 7440  2_𝑜c2o 7441  3_𝑜c3o 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-1o 7447  df-2o 7448  df-3o 7449

This theorem is referenced by:  clsk1independent  37364