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Theorem clsk1indlem4 38868
 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 7104 . . . . . . . . . 10 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4482 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5 0ex 4924 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4451 . . . . . . . . . . 11 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
74, 6sylibr 224 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8 snsstp2 4483 . . . . . . . . . . . 12 {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10 1oex 7721 . . . . . . . . . . . 12 1𝑜 ∈ V
1110snss 4451 . . . . . . . . . . 11 (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
129, 11sylibr 224 . . . . . . . . . 10 (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
137, 12prssd 4488 . . . . . . . . 9 (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
142, 13sselpwd 4941 . . . . . . . 8 (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
1514trud 1641 . . . . . . 7 {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
16 df3o2 38848 . . . . . . . 8 3𝑜 = {∅, 1𝑜, 2𝑜}
1716pweqi 4301 . . . . . . 7 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
1815, 17eleqtrri 2849 . . . . . 6 {∅, 1𝑜} ∈ 𝒫 3𝑜
1918a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3𝑜 → {∅, 1𝑜} ∈ 𝒫 3𝑜)
20 id 22 . . . . 5 (𝑠 ∈ 𝒫 3𝑜𝑠 ∈ 𝒫 3𝑜)
2119, 20ifcld 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)))
2523, 24bitri 264 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2622, 25syl6bb 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))
2826, 27ifbieq2d 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 {∅} = {∅, ∅}
3130eqeq1i 2776 . . . . . . . . . . 11 ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
325a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
33 1on 7720 . . . . . . . . . . . . . 14 1𝑜 ∈ On
3433a1i 11 . . . . . . . . . . . . 13 (⊤ → 1𝑜 ∈ On)
3532, 34preq2b 4510 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
3635trud 1641 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37 eqcom 2778 . . . . . . . . . . 11 (∅ = 1𝑜 ↔ 1𝑜 = ∅)
3831, 36, 373bitri 286 . . . . . . . . . 10 ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
3929, 38nemtbir 3038 . . . . . . . . 9 ¬ {∅} = {∅, 1𝑜}
4039intnan 474 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41 pm3.24 389 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2778 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 611 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 311 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 865 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4235 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
4728, 46syl6eq 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
5149, 50ifex 4295 . . . . 5 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
5247, 48, 51fvmpt 6424 . . . 4 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5321, 52syl 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 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4250 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5756, 48, 51fvmpt 6424 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5857fveq2d 6336 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
5953, 58, 573eqtr4d 2815 . 2 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 3071 1 𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
 Colors of variables: wff setvar class 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  1𝑜c1o 7706  2𝑜c2o 7707  3𝑜c3o 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
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