Theorem clsk1indlem4 40401

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 7472	. . . . . . . . . 10 ⊢ {∅, 1o, 2o} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
3		snsstp1 4751	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1o, 2o}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
5		0ex 5213	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4720	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
7	4, 6	sylibr 236	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
8		snsstp2 4752	. . . . . . . . . . . 12 ⊢ {1o} ⊆ {∅, 1o, 2o}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1o} ⊆ {∅, 1o, 2o})
10		1oex 8112	. . . . . . . . . . . 12 ⊢ 1o ∈ V
11	10	snss 4720	. . . . . . . . . . 11 ⊢ (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
12	9, 11	sylibr 236	. . . . . . . . . 10 ⊢ (⊤ → 1o ∈ {∅, 1o, 2o})
13	7, 12	prssd 4757	. . . . . . . . 9 ⊢ (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
14	2, 13	sselpwd 5232	. . . . . . . 8 ⊢ (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
15	14	mptru 1544	. . . . . . 7 ⊢ {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16		df3o2 40381	. . . . . . . 8 ⊢ 3o = {∅, 1o, 2o}
17	16	pweqi 4559	. . . . . . 7 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
18	15, 17	eleqtrri 2914	. . . . . 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 4514	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22		eqeq1 2827	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23		eqcom 2830	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24		eqif 4509	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
25	23, 24	bitri 277	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	22, 25	syl6bb 289	. . . . . . 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 4494	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29		1n0 8121	. . . . . . . . . 10 ⊢ 1o ≠ ∅
30		dfsn2 4582	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
31	30	eqeq1i 2828	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
32	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
33		1on 8111	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1o ∈ On)
35	32, 34	preq2b 4780	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
36	35	mptru 1544	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37		eqcom 2830	. . . . . . . . . . 11 ⊢ (∅ = 1o ↔ 1o = ∅)
38	31, 36, 37	3bitri 299	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1o} ↔ 1o = ∅)
39	29, 38	nemtbir 3114	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1o}
40	39	intnan 489	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41		pm3.24 405	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2830	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 626	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 324	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 877	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4479	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
47	28, 46	syl6eq 2874	. . . . 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 5335	. . . . . 6 ⊢ {∅, 1o} ∈ V
50		vex 3499	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4517	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 6770	. . . 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 2827	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 22	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4494	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
57	56, 48, 51	fvmpt 6770	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
58	57	fveq2d 6676	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
59	53, 58, 57	3eqtr4d 2868	. 2 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 3150	1 ⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ⊤wtru 1538   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  ifcif 4469  𝒫 cpw 4541  {csn 4569  {cpr 4571  {ctp 4573   ↦ cmpt 5148  Oncon0 6193  ‘cfv 6357  1_oc1o 8097  2_oc2o 8098  3_oc3o 8099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fv 6365  df-1o 8104  df-2o 8105  df-3o 8106

This theorem is referenced by:  clsk1independent  40403