MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmphindis Structured version   Visualization version   GIF version

Theorem hmphindis 22409
Description: Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmphdis.1 𝑋 = 𝐽
Assertion
Ref Expression
hmphindis (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})

Proof of Theorem hmphindis
StepHypRef Expression
1 dfsn2 4563 . . 3 {∅} = {∅, ∅}
2 indislem 21612 . . . . . . 7 {∅, ( I ‘𝐴)} = {∅, 𝐴}
3 preq2 4655 . . . . . . . 8 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅, ∅})
43, 1eqtr4di 2877 . . . . . . 7 (( I ‘𝐴) = ∅ → {∅, ( I ‘𝐴)} = {∅})
52, 4syl5eqr 2873 . . . . . 6 (( I ‘𝐴) = ∅ → {∅, 𝐴} = {∅})
65breq2d 5064 . . . . 5 (( I ‘𝐴) = ∅ → (𝐽 ≃ {∅, 𝐴} ↔ 𝐽 ≃ {∅}))
76biimpac 482 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 ≃ {∅})
8 hmph0 22407 . . . 4 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
97, 8sylib 221 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
109unieqd 4838 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅})
11 hmphdis.1 . . . . 5 𝑋 = 𝐽
12 0ex 5197 . . . . . . 7 ∅ ∈ V
1312unisn 4844 . . . . . 6 {∅} = ∅
1413eqcomi 2833 . . . . 5 ∅ = {∅}
1510, 11, 143eqtr4g 2884 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝑋 = ∅)
1615preq2d 4661 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → {∅, 𝑋} = {∅, ∅})
171, 9, 163eqtr4a 2885 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) = ∅) → 𝐽 = {∅, 𝑋})
18 hmphen 22397 . . . . 5 (𝐽 ≃ {∅, 𝐴} → 𝐽 ≈ {∅, 𝐴})
19 necom 3067 . . . . . . . 8 (( I ‘𝐴) ≠ ∅ ↔ ∅ ≠ ( I ‘𝐴))
20 fvex 6674 . . . . . . . . 9 ( I ‘𝐴) ∈ V
21 pr2nelem 9428 . . . . . . . . 9 ((∅ ∈ V ∧ ( I ‘𝐴) ∈ V ∧ ∅ ≠ ( I ‘𝐴)) → {∅, ( I ‘𝐴)} ≈ 2o)
2212, 20, 21mp3an12 1448 . . . . . . . 8 (∅ ≠ ( I ‘𝐴) → {∅, ( I ‘𝐴)} ≈ 2o)
2319, 22sylbi 220 . . . . . . 7 (( I ‘𝐴) ≠ ∅ → {∅, ( I ‘𝐴)} ≈ 2o)
2423adantl 485 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, ( I ‘𝐴)} ≈ 2o)
252, 24eqbrtrrid 5088 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → {∅, 𝐴} ≈ 2o)
26 entr 8557 . . . . 5 ((𝐽 ≈ {∅, 𝐴} ∧ {∅, 𝐴} ≈ 2o) → 𝐽 ≈ 2o)
2718, 25, 26syl2an2r 684 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ≈ 2o)
28 hmphtop1 22391 . . . . . . 7 (𝐽 ≃ {∅, 𝐴} → 𝐽 ∈ Top)
2928adantr 484 . . . . . 6 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ Top)
3011toptopon 21529 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3129, 30sylib 221 . . . . 5 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 ∈ (TopOn‘𝑋))
32 en2top 21597 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3331, 32syl 17 . . . 4 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 ≈ 2o ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))
3427, 33mpbid 235 . . 3 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅))
3534simpld 498 . 2 ((𝐽 ≃ {∅, 𝐴} ∧ ( I ‘𝐴) ≠ ∅) → 𝐽 = {∅, 𝑋})
3617, 35pm2.61dane 3101 1 (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  c0 4276  {csn 4550  {cpr 4552   cuni 4824   class class class wbr 5052   I cid 5446  cfv 6343  2oc2o 8092  cen 8502  Topctop 21505  TopOnctopon 21522  chmph 22366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-1o 8098  df-2o 8099  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-card 9365  df-top 21506  df-topon 21523  df-cn 21839  df-hmeo 22367  df-hmph 22368
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator