Theorem clsk1indlem4 38868

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 7104	. . . . . . . . . 10 ⊢ {∅, 1𝑜, 2𝑜} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3		snsstp1 4482	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1𝑜, 2𝑜}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5		0ex 4924	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4451	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
7	4, 6	sylibr 224	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8		snsstp2 4483	. . . . . . . . . . . 12 ⊢ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10		1oex 7721	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
11	10	snss 4451	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
12	9, 11	sylibr 224	. . . . . . . . . 10 ⊢ (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
13	7, 12	prssd 4488	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
14	2, 13	sselpwd 4941	. . . . . . . 8 ⊢ (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
15	14	trud 1641	. . . . . . 7 ⊢ {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
16		df3o2 38848	. . . . . . . 8 ⊢ 3𝑜 = {∅, 1𝑜, 2𝑜}
17	16	pweqi 4301	. . . . . . 7 ⊢ 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
18	15, 17	eleqtrri 2849	. . . . . 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 4270	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
22		eqeq1 2775	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
23		eqcom 2778	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
24		eqif 4265	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
25	23, 24	bitri 264	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	22, 25	syl6bb 276	. . . . . . 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 4250	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
29		1n0 7729	. . . . . . . . . 10 ⊢ 1𝑜 ≠ ∅
30		dfsn2 4329	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
31	30	eqeq1i 2776	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
32	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
33		1on 7720	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1𝑜 ∈ On)
35	32, 34	preq2b 4510	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
36	35	trud 1641	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37		eqcom 2778	. . . . . . . . . . 11 ⊢ (∅ = 1𝑜 ↔ 1𝑜 = ∅)
38	31, 36, 37	3bitri 286	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
39	29, 38	nemtbir 3038	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1𝑜}
40	39	intnan 474	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41		pm3.24 389	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2778	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 611	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 311	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 865	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4235	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
47	28, 46	syl6eq 2821	. . . . 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 5037	. . . . . 6 ⊢ {∅, 1𝑜} ∈ V
50		vex 3354	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4295	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 6424	. . . 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 2775	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 22	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4250	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
57	56, 48, 51	fvmpt 6424	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
58	57	fveq2d 6336	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
59	53, 58, 57	3eqtr4d 2815	. 2 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 3071	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 382   ∨ wo 834   = wceq 1631  ⊤wtru 1632   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ⊆ wss 3723  ∅c0 4063  ifcif 4225  𝒫 cpw 4297  {csn 4316  {cpr 4318  {ctp 4320   ↦ cmpt 4863  Oncon0 5866  ‘cfv 6031  1_𝑜c1o 7706  2_𝑜c2o 7707  3_𝑜c3o 7708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-1o 7713  df-2o 7714  df-3o 7715

This theorem is referenced by:  clsk1independent  38870