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 38659
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 6999 . . . . . . . . . 10 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . . . . . 9 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4379 . . . . . . . . . . . 12 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . . . . . . . 11 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5 0ex 4823 . . . . . . . . . . . 12 ∅ ∈ V
65snss 4348 . . . . . . . . . . 11 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
74, 6sylibr 224 . . . . . . . . . 10 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
8 snsstp2 4380 . . . . . . . . . . . 12 {1𝑜} ⊆ {∅, 1𝑜, 2𝑜}
98a1i 11 . . . . . . . . . . 11 (⊤ → {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
10 1on 7612 . . . . . . . . . . . . 13 1𝑜 ∈ On
1110elexi 3244 . . . . . . . . . . . 12 1𝑜 ∈ V
1211snss 4348 . . . . . . . . . . 11 (1𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
139, 12sylibr 224 . . . . . . . . . 10 (⊤ → 1𝑜 ∈ {∅, 1𝑜, 2𝑜})
147, 13prssd 4386 . . . . . . . . 9 (⊤ → {∅, 1𝑜} ⊆ {∅, 1𝑜, 2𝑜})
152, 14sselpwd 4840 . . . . . . . 8 (⊤ → {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
1615trud 1533 . . . . . . 7 {∅, 1𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
17 df3o2 38639 . . . . . . . 8 3𝑜 = {∅, 1𝑜, 2𝑜}
1817pweqi 4195 . . . . . . 7 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
1916, 18eleqtrri 2729 . . . . . 6 {∅, 1𝑜} ∈ 𝒫 3𝑜
2019a1i 11 . . . . 5 (𝑠 ∈ 𝒫 3𝑜 → {∅, 1𝑜} ∈ 𝒫 3𝑜)
21 id 22 . . . . 5 (𝑠 ∈ 𝒫 3𝑜𝑠 ∈ 𝒫 3𝑜)
2220, 21ifcld 4164 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜)
23 eqeq1 2655 . . . . . . . 8 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅}))
24 eqcom 2658 . . . . . . . . 9 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ {∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
25 eqif 4159 . . . . . . . . 9 ({∅} = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2624, 25bitri 264 . . . . . . . 8 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)))
2723, 26syl6bb 276 . . . . . . 7 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → (𝑟 = {∅} ↔ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))))
28 id 22 . . . . . . 7 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → 𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
2927, 28ifbieq2d 4144 . . . . . 6 (𝑟 = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
30 1n0 7620 . . . . . . . . . 10 1𝑜 ≠ ∅
31 dfsn2 4223 . . . . . . . . . . . 12 {∅} = {∅, ∅}
3231eqeq1i 2656 . . . . . . . . . . 11 ({∅} = {∅, 1𝑜} ↔ {∅, ∅} = {∅, 1𝑜})
335a1i 11 . . . . . . . . . . . . 13 (⊤ → ∅ ∈ V)
3410a1i 11 . . . . . . . . . . . . 13 (⊤ → 1𝑜 ∈ On)
3533, 34preq2b 4410 . . . . . . . . . . . 12 (⊤ → ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜))
3635trud 1533 . . . . . . . . . . 11 ({∅, ∅} = {∅, 1𝑜} ↔ ∅ = 1𝑜)
37 eqcom 2658 . . . . . . . . . . 11 (∅ = 1𝑜 ↔ 1𝑜 = ∅)
3832, 36, 373bitri 286 . . . . . . . . . 10 ({∅} = {∅, 1𝑜} ↔ 1𝑜 = ∅)
3930, 38nemtbir 2918 . . . . . . . . 9 ¬ {∅} = {∅, 1𝑜}
4039intnan 980 . . . . . . . 8 ¬ (𝑠 = {∅} ∧ {∅} = {∅, 1𝑜})
41 pm3.24 944 . . . . . . . . 9 ¬ (𝑠 = {∅} ∧ ¬ 𝑠 = {∅})
42 eqcom 2658 . . . . . . . . . 10 (𝑠 = {∅} ↔ {∅} = 𝑠)
4342anbi2ci 732 . . . . . . . . 9 ((𝑠 = {∅} ∧ ¬ 𝑠 = {∅}) ↔ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4441, 43mtbi 311 . . . . . . . 8 ¬ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)
4540, 44pm3.2ni 917 . . . . . . 7 ¬ ((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠))
4645iffalsei 4129 . . . . . 6 if(((𝑠 = {∅} ∧ {∅} = {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ {∅} = 𝑠)), {∅, 1𝑜}, if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)
4729, 46syl6eq 2701 . . . . 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 4939 . . . . . 6 {∅, 1𝑜} ∈ V
50 vex 3234 . . . . . 6 𝑠 ∈ V
5149, 50ifex 4189 . . . . 5 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
5247, 48, 51fvmpt 6321 . . . 4 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5322, 52syl 17 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
54 eqeq1 2655 . . . . . 6 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
55 id 22 . . . . . 6 (𝑟 = 𝑠𝑟 = 𝑠)
5654, 55ifbieq2d 4144 . . . . 5 (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5756, 48, 51fvmpt 6321 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
5857fveq2d 6233 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾‘if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠)))
5953, 58, 573eqtr4d 2695 . 2 (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘(𝐾𝑠)) = (𝐾𝑠))
6059rgen 2951 1 𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾𝑠)) = (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1523  wtru 1524  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210  {cpr 4212  {ctp 4214  cmpt 4762  Oncon0 5761  cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599  3𝑜c3o 7600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-1o 7605  df-2o 7606  df-3o 7607
This theorem is referenced by:  clsk1independent  38661
  Copyright terms: Public domain W3C validator