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Theorem clsk1indlem4 41678
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 7617 . . . . . . . . . 10 {∅, 1o, 2o} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1o, 2o} ∈ V)
3 snsstp1 4752 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1o, 2o}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1o, 2o})
5 0ex 5234 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4722 . . . . . . . . . . 11 (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
74, 6sylibr 233 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1o, 2o})
8 snsstp2 4753 . . . . . . . . . . . 12 {1o} ⊆ {∅, 1o, 2o}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1o} ⊆ {∅, 1o, 2o})
10 1oex 8327 . . . . . . . . . . . 12 1o ∈ V
1110snss 4722 . . . . . . . . . . 11 (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
129, 11sylibr 233 . . . . . . . . . 10 (⊤ → 1o ∈ {∅, 1o, 2o})
137, 12prssd 4758 . . . . . . . . 9 (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
142, 13sselpwd 5253 . . . . . . . 8 (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
1514mptru 1544 . . . . . . 7 {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16 df3o2 41658 . . . . . . . 8 3o = {∅, 1o, 2o}
1716pweqi 4554 . . . . . . 7 𝒫 3o = 𝒫 {∅, 1o, 2o}
1815, 17eleqtrri 2833 . . . . . 6 {∅, 1o} ∈ 𝒫 3o
1918a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3o → {∅, 1o} ∈ 𝒫 3o)
20 id 22 . . . . 5 (𝑠 ∈ 𝒫 3o𝑠 ∈ 𝒫 3o)
2119, 20ifcld 4508 . . . 4 (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22 eqeq1 2737 . . . . . . . 8 (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23 eqcom 2740 . . . . . . . . 9 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24 eqif 4503 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2523, 24bitri 274 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2622, 25bitrdi 286 . . . . . . 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 4488 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29 1n0 8338 . . . . . . . . . 10 1o ≠ ∅
30 dfsn2 4577 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3130eqeq1i 2738 . . . . . . . . . . 11 ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
325a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
33 1on 8329 . . . . . . . . . . . . . 14 1o ∈ On
3433a1i 11 . . . . . . . . . . . . 13 (⊤ → 1o ∈ On)
3532, 34preq2b 4781 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
3635mptru 1544 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37 eqcom 2740 . . . . . . . . . . 11 (∅ = 1o ↔ 1o = ∅)
3831, 36, 373bitri 296 . . . . . . . . . 10 ({∅} = {∅, 1o} ↔ 1o = ∅)
3929, 38nemtbir 3035 . . . . . . . . 9 ¬ {∅} = {∅, 1o}
4039intnan 486 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41 pm3.24 402 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2740 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 624 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 321 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 877 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4472 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
4728, 46eqtrdi 2789 . . . . 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 5358 . . . . . 6 {∅, 1o} ∈ V
50 vex 3438 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4512 . . . . 5 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
5247, 48, 51fvmpt 6895 . . . 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 2737 . . . . . 6 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55 id 22 . . . . . 6 (𝑟 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4488 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5756, 48, 51fvmpt 6895 . . . 4 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5857fveq2d 6796 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
5953, 58, 573eqtr4d 2783 . 2 (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 3061 1 𝑠 ∈ 𝒫 3o(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 843   = wceq 1537  wtru 1538  wcel 2101  wral 3059  Vcvv 3434  wss 3889  c0 4259  ifcif 4462  𝒫 cpw 4536  {csn 4564  {cpr 4566  {ctp 4568  cmpt 5160  Oncon0 6270  cfv 6447  1oc1o 8310  2oc2o 8311  3oc3o 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-ord 6273  df-on 6274  df-suc 6276  df-iota 6399  df-fun 6449  df-fv 6455  df-1o 8317  df-2o 8318  df-3o 8319
This theorem is referenced by:  clsk1independent  41680
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