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Theorem clsk1indlem4 38882
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 7125 . . . . . . . . . 10 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4493 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5 0ex 4937 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4462 . . . . . . . . . . 11 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
74, 6sylibr 225 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8 snsstp2 4494 . . . . . . . . . . . 12 {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10 1oex 7742 . . . . . . . . . . . 12 1𝑜 ∈ V
1110snss 4462 . . . . . . . . . . 11 (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
129, 11sylibr 225 . . . . . . . . . 10 (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
137, 12prssd 4499 . . . . . . . . 9 (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
142, 13sselpwd 4955 . . . . . . . 8 (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
1514trud 1644 . . . . . . 7 {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
16 df3o2 38862 . . . . . . . 8 3𝑜 = {∅, 1𝑜, 2𝑜}
1716pweqi 4311 . . . . . . 7 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
1815, 17eleqtrri 2852 . . . . . 6 {∅, 1𝑜} ∈ 𝒫 3𝑜
1918a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3𝑜 → {∅, 1𝑜} ∈ 𝒫 3𝑜)
20 id 22 . . . . 5 (𝑠 ∈ 𝒫 3𝑜𝑠 ∈ 𝒫 3𝑜)
2119, 20ifcld 4280 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
22 eqeq1 2778 . . . . . . . 8 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
23 eqcom 2781 . . . . . . . . 9 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
24 eqif 4275 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2523, 24bitri 265 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2622, 25syl6bb 277 . . . . . . 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 4260 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
29 1n0 7750 . . . . . . . . . 10 1𝑜 ≠ ∅
30 dfsn2 4339 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3130eqeq1i 2779 . . . . . . . . . . 11 ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
325a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
33 1on 7741 . . . . . . . . . . . . . 14 1𝑜 ∈ On
3433a1i 11 . . . . . . . . . . . . 13 (⊤ → 1𝑜 ∈ On)
3532, 34preq2b 4521 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
3635trud 1644 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37 eqcom 2781 . . . . . . . . . . 11 (∅ = 1𝑜 ↔ 1𝑜 = ∅)
3831, 36, 373bitri 287 . . . . . . . . . 10 ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
3929, 38nemtbir 3041 . . . . . . . . 9 ¬ {∅} = {∅, 1𝑜}
4039intnan 475 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41 pm3.24 390 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2781 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 612 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 312 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 894 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4245 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
4728, 46syl6eq 2824 . . . . 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 5051 . . . . . 6 {∅, 1𝑜} ∈ V
50 vex 3358 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4305 . . . . 5 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
5247, 48, 51fvmpt 6441 . . . 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 2778 . . . . . 6 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55 id 22 . . . . . 6 (𝑟 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4260 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5756, 48, 51fvmpt 6441 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5857fveq2d 6352 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
5953, 58, 573eqtr4d 2818 . 2 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 3074 1 𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 383  wo 863   = wceq 1634  wtru 1635  wcel 2148  wral 3064  Vcvv 3355  wss 3729  c0 4073  ifcif 4235  𝒫 cpw 4307  {csn 4326  {cpr 4328  {ctp 4330  cmpt 4876  Oncon0 5877  cfv 6042  1𝑜c1o 7727  2𝑜c2o 7728  3𝑜c3o 7729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-ord 5880  df-on 5881  df-suc 5883  df-iota 6005  df-fun 6044  df-fv 6050  df-1o 7734  df-2o 7735  df-3o 7736
This theorem is referenced by:  clsk1independent  38884
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