Theorem clsk1indlem4 38342

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 6957	. . . . . . . . . 10 ⊢ {∅, 1𝑜, 2𝑜} ∈ V
2	1	a1i 11	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3		snsstp1 4347	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, 1𝑜, 2𝑜}
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5		0ex 4790	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	snss 4316	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
7	4, 6	sylibr 224	. . . . . . . . . 10 ⊢ (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8		snsstp2 4348	. . . . . . . . . . . 12 ⊢ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10		1on 7567	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ On
11	10	elexi 3213	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
12	11	snss 4316	. . . . . . . . . . 11 ⊢ (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
13	9, 12	sylibr 224	. . . . . . . . . 10 ⊢ (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
14	7, 13	prssd 4354	. . . . . . . . 9 ⊢ (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15	2, 14	sselpwd 4807	. . . . . . . 8 ⊢ (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
16	15	trud 1493	. . . . . . 7 ⊢ {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
17		df3o2 38322	. . . . . . . 8 ⊢ 3𝑜 = {∅, 1𝑜, 2𝑜}
18	17	pweqi 4162	. . . . . . 7 ⊢ 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
19	16, 18	eleqtrri 2700	. . . . . 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 4131	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
23		eqeq1 2626	. . . . . . . 8 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
24		eqcom 2629	. . . . . . . . 9 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
25		eqif 4126	. . . . . . . . 9 ⊢ ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
26	24, 25	bitri 264	. . . . . . . 8 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
27	23, 26	syl6bb 276	. . . . . . 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 4111	. . . . . 6 ⊢ (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
30		1n0 7575	. . . . . . . . . 10 ⊢ 1𝑜 ≠ ∅
31		dfsn2 4190	. . . . . . . . . . . 12 ⊢ {∅} = {∅, ∅}
32	31	eqeq1i 2627	. . . . . . . . . . 11 ⊢ ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
33	5	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ ∈ V)
34	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1𝑜 ∈ On)
35	33, 34	preq2b 4378	. . . . . . . . . . . 12 ⊢ (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
36	35	trud 1493	. . . . . . . . . . 11 ⊢ ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37		eqcom 2629	. . . . . . . . . . 11 ⊢ (∅ = 1𝑜 ↔ 1𝑜 = ∅)
38	32, 36, 37	3bitri 286	. . . . . . . . . 10 ⊢ ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
39	30, 38	nemtbir 2889	. . . . . . . . 9 ⊢ ¬ {∅} = {∅, 1𝑜}
40	39	intnan 960	. . . . . . . 8 ⊢ ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41		pm3.24 926	. . . . . . . . 9 ⊢ ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42		eqcom 2629	. . . . . . . . . 10 ⊢ (𝑠 = {∅} ↔ {∅} = 𝑠)
43	42	anbi2ci 732	. . . . . . . . 9 ⊢ ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
44	41, 43	mtbi 312	. . . . . . . 8 ⊢ ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
45	40, 44	pm3.2ni 899	. . . . . . 7 ⊢ ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
46	45	iffalsei 4096	. . . . . 6 ⊢ if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
47	29, 46	syl6eq 2672	. . . . 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 4909	. . . . . 6 ⊢ {∅, 1𝑜} ∈ V
50		vex 3203	. . . . . 6 ⊢ 𝑠 ∈ V
51	49, 50	ifex 4156	. . . . 5 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
52	47, 48, 51	fvmpt 6282	. . . 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 2626	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55		id 22	. . . . . 6 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
56	54, 55	ifbieq2d 4111	. . . . 5 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
57	56, 48, 51	fvmpt 6282	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
58	57	fveq2d 6195	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
59	53, 58, 57	3eqtr4d 2666	. 2 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))
60	59	rgen 2922	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179  {ctp 4181   ↦ cmpt 4729  Oncon0 5723  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554  3_𝑜c3o 7555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by:  clsk1independent  38344