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Theorem clsk1indlem4 41334
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 7532 . . . . . . . . . 10 {∅, 1o, 2o} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1o, 2o} ∈ V)
3 snsstp1 4729 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1o, 2o}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1o, 2o})
5 0ex 5200 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4699 . . . . . . . . . . 11 (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
74, 6sylibr 237 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1o, 2o})
8 snsstp2 4730 . . . . . . . . . . . 12 {1o} ⊆ {∅, 1o, 2o}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1o} ⊆ {∅, 1o, 2o})
10 1oex 8215 . . . . . . . . . . . 12 1o ∈ V
1110snss 4699 . . . . . . . . . . 11 (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
129, 11sylibr 237 . . . . . . . . . 10 (⊤ → 1o ∈ {∅, 1o, 2o})
137, 12prssd 4735 . . . . . . . . 9 (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
142, 13sselpwd 5219 . . . . . . . 8 (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
1514mptru 1550 . . . . . . 7 {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16 df3o2 41314 . . . . . . . 8 3o = {∅, 1o, 2o}
1716pweqi 4531 . . . . . . 7 𝒫 3o = 𝒫 {∅, 1o, 2o}
1815, 17eleqtrri 2837 . . . . . 6 {∅, 1o} ∈ 𝒫 3o
1918a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3o → {∅, 1o} ∈ 𝒫 3o)
20 id 22 . . . . 5 (𝑠 ∈ 𝒫 3o𝑠 ∈ 𝒫 3o)
2119, 20ifcld 4485 . . . 4 (𝑠 ∈ 𝒫 3o → if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ 𝒫 3o)
22 eqeq1 2741 . . . . . . . 8 (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅}))
23 eqcom 2744 . . . . . . . . 9 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
24 eqif 4480 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2523, 24bitri 278 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2622, 25bitrdi 290 . . . . . . 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 4465 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29 1n0 8221 . . . . . . . . . 10 1o ≠ ∅
30 dfsn2 4554 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3130eqeq1i 2742 . . . . . . . . . . 11 ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
325a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
33 1on 8209 . . . . . . . . . . . . . 14 1o ∈ On
3433a1i 11 . . . . . . . . . . . . 13 (⊤ → 1o ∈ On)
3532, 34preq2b 4758 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
3635mptru 1550 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37 eqcom 2744 . . . . . . . . . . 11 (∅ = 1o ↔ 1o = ∅)
3831, 36, 373bitri 300 . . . . . . . . . 10 ({∅} = {∅, 1o} ↔ 1o = ∅)
3929, 38nemtbir 3037 . . . . . . . . 9 ¬ {∅} = {∅, 1o}
4039intnan 490 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41 pm3.24 406 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2744 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 628 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 325 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 881 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4449 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)
4728, 46eqtrdi 2794 . . . . 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 5325 . . . . . 6 {∅, 1o} ∈ V
50 vex 3412 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4489 . . . . 5 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
5247, 48, 51fvmpt 6818 . . . 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 2741 . . . . . 6 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55 id 22 . . . . . 6 (𝑟 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4465 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5756, 48, 51fvmpt 6818 . . . 4 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5857fveq2d 6721 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
5953, 58, 573eqtr4d 2787 . 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 209  wa 399  wo 847   = wceq 1543  wtru 1544  wcel 2110  wral 3061  Vcvv 3408  wss 3866  c0 4237  ifcif 4439  𝒫 cpw 4513  {csn 4541  {cpr 4543  {ctp 4545  cmpt 5135  Oncon0 6213  cfv 6380  1oc1o 8195  2oc2o 8196  3oc3o 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fv 6388  df-1o 8202  df-2o 8203  df-3o 8204
This theorem is referenced by:  clsk1independent  41336
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