Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clsk1indlem4 Structured version   Visualization version   GIF version

Theorem clsk1indlem4 42404
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 7682 . . . . . . . . . 10 {∅, 1o, 2o} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1o, 2o} ∈ V)
3 snsstp1 4777 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1o, 2o}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1o, 2o})
5 0ex 5265 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4747 . . . . . . . . . . 11 (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
74, 6sylibr 233 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1o, 2o})
8 snsstp2 4778 . . . . . . . . . . . 12 {1o} ⊆ {∅, 1o, 2o}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1o} ⊆ {∅, 1o, 2o})
10 1oex 8423 . . . . . . . . . . . 12 1o ∈ V
1110snss 4747 . . . . . . . . . . 11 (1o ∈ {∅, 1o, 2o} ↔ {1o} ⊆ {∅, 1o, 2o})
129, 11sylibr 233 . . . . . . . . . 10 (⊤ → 1o ∈ {∅, 1o, 2o})
137, 12prssd 4783 . . . . . . . . 9 (⊤ → {∅, 1o} ⊆ {∅, 1o, 2o})
142, 13sselpwd 5284 . . . . . . . 8 (⊤ → {∅, 1o} ∈ 𝒫 {∅, 1o, 2o})
1514mptru 1549 . . . . . . 7 {∅, 1o} ∈ 𝒫 {∅, 1o, 2o}
16 df3o2 41691 . . . . . . . 8 3o = {∅, 1o, 2o}
1716pweqi 4577 . . . . . . 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 4533 . . . 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 4528 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2523, 24bitri 275 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2622, 25bitrdi 287 . . . . . . 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 4513 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1o}, 𝑠) → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1o}, if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
29 1n0 8435 . . . . . . . . . 10 1o ≠ ∅
30 dfsn2 4600 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3130eqeq1i 2738 . . . . . . . . . . 11 ({∅} = {∅, 1o} ↔ {∅, ∅} = {∅, 1o})
325a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
33 1on 8425 . . . . . . . . . . . . . 14 1o ∈ On
3433a1i 11 . . . . . . . . . . . . 13 (⊤ → 1o ∈ On)
3532, 34preq2b 4806 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o))
3635mptru 1549 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1o} ↔ ∅ = 1o)
37 eqcom 2740 . . . . . . . . . . 11 (∅ = 1o ↔ 1o = ∅)
3831, 36, 373bitri 297 . . . . . . . . . 10 ({∅} = {∅, 1o} ↔ 1o = ∅)
3929, 38nemtbir 3037 . . . . . . . . 9 ¬ {∅} = {∅, 1o}
4039intnan 488 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1o})
41 pm3.24 404 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2740 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 626 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 322 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 880 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4497 . . . . . 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 5390 . . . . . 6 {∅, 1o} ∈ V
50 vex 3448 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4537 . . . . 5 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
5247, 48, 51fvmpt 6949 . . . 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 4513 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5756, 48, 51fvmpt 6949 . . . 4 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
5857fveq2d 6847 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1o}, 𝑠)))
5953, 58, 573eqtr4d 2783 . 2 (𝑠 ∈ 𝒫 3o → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 3063 1 𝑠 ∈ 𝒫 3o(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846   = wceq 1542  wtru 1543  wcel 2107  wral 3061  Vcvv 3444  wss 3911  c0 4283  ifcif 4487  𝒫 cpw 4561  {csn 4587  {cpr 4589  {ctp 4591  cmpt 5189  Oncon0 6318  cfv 6497  1oc1o 8406  2oc2o 8407  3oc3o 8408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fv 6505  df-1o 8413  df-2o 8414  df-3o 8415
This theorem is referenced by:  clsk1independent  42406
  Copyright terms: Public domain W3C validator