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Theorem clsk1indlem4 37821
 Description: The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 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‘(𝐾𝑠)) = (𝐾𝑠)
Distinct variable group:   𝑠,𝑟
Allowed substitution hints:   𝐾(𝑠,𝑟)

Proof of Theorem clsk1indlem4
StepHypRef Expression
1 tpex 6910 . . . . . . . . . 10 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4315 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5 0ex 4750 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4286 . . . . . . . . . . 11 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
74, 6sylibr 224 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8 snsstp2 4316 . . . . . . . . . . . 12 {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10 1on 7512 . . . . . . . . . . . . 13 1𝑜 ∈ On
1110elexi 3199 . . . . . . . . . . . 12 1𝑜 ∈ V
1211snss 4286 . . . . . . . . . . 11 (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
139, 12sylibr 224 . . . . . . . . . 10 (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
147, 13prssd 4322 . . . . . . . . 9 (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
152, 14sselpwd 4767 . . . . . . . 8 (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
1615trud 1490 . . . . . . 7 {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
17 df3o2 37801 . . . . . . . 8 3𝑜 = {∅, 1𝑜, 2𝑜}
1817pweqi 4134 . . . . . . 7 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
1916, 18eleqtrri 2697 . . . . . 6 {∅, 1𝑜} ∈ 𝒫 3𝑜
2019a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3𝑜 → {∅, 1𝑜} ∈ 𝒫 3𝑜)
21 id 22 . . . . 5 (𝑠 ∈ 𝒫 3𝑜𝑠 ∈ 𝒫 3𝑜)
2220, 21ifcld 4103 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
23 eqeq1 2625 . . . . . . . 8 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
24 eqcom 2628 . . . . . . . . 9 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
25 eqif 4098 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2624, 25bitri 264 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2723, 26syl6bb 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))
2927, 28ifbieq2d 4083 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
30 1n0 7520 . . . . . . . . . 10 1𝑜 ≠ ∅
31 dfsn2 4161 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3231eqeq1i 2626 . . . . . . . . . . 11 ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
335a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
3410a1i 11 . . . . . . . . . . . . 13 (⊤ → 1𝑜 ∈ On)
3533, 34preq2b 4346 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
3635trud 1490 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37 eqcom 2628 . . . . . . . . . . 11 (∅ = 1𝑜 ↔ 1𝑜 = ∅)
3832, 36, 373bitri 286 . . . . . . . . . 10 ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
3930, 38nemtbir 2885 . . . . . . . . 9 ¬ {∅} = {∅, 1𝑜}
4039intnan 959 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41 pm3.24 925 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2628 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 731 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 312 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 898 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4068 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
4729, 46syl6eq 2671 . . . . 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 4870 . . . . . 6 {∅, 1𝑜} ∈ V
50 vex 3189 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4128 . . . . 5 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
5247, 48, 51fvmpt 6239 . . . 4 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5322, 52syl 17 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
54 eqeq1 2625 . . . . . 6 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55 id 22 . . . . . 6 (𝑟 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4083 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5756, 48, 51fvmpt 6239 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5857fveq2d 6152 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
5953, 58, 573eqtr4d 2665 . 2 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 2917 1 𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480  ⊤wtru 1481   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ⊆ wss 3555  ∅c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148  {cpr 4150  {ctp 4152   ↦ cmpt 4673  Oncon0 5682  ‘cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499  3𝑜c3o 7500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-1o 7505  df-2o 7506  df-3o 7507 This theorem is referenced by:  clsk1independent  37823
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