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Theorem indishmph 21511
Description: Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
indishmph (𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})

Proof of Theorem indishmph
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 7908 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 6094 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 f1odm 6098 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
4 vex 3189 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7046 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5syl6eqelr 2707 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
7 f1ofo 6101 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
8 fornex 7082 . . . . . . . . 9 (𝐴 ∈ V → (𝑓:𝐴onto𝐵𝐵 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
109, 6elmapd 7816 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
112, 10mpbird 247 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐵𝑚 𝐴))
12 indistopon 20715 . . . . . . . 8 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14 cnindis 21006 . . . . . . 7 (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1513, 9, 14syl2anc 692 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1611, 15eleqtrrd 2701 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17 f1ocnv 6106 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
18 f1of 6094 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵𝐴)
206, 9elmapd 7816 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
2119, 20mpbird 247 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐴𝑚 𝐵))
22 indistopon 20715 . . . . . . . 8 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24 cnindis 21006 . . . . . . 7 (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2523, 6, 24syl2anc 692 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2621, 25eleqtrrd 2701 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27 ishmeo 21472 . . . . 5 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ 𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
2816, 26, 27sylanbrc 697 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29 hmphi 21490 . . . 4 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
3028, 29syl 17 . . 3 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
3130exlimiv 1855 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
321, 31sylbi 207 1 (𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  c0 3891  {cpr 4150   class class class wbr 4613  ccnv 5073  dom cdm 5074  wf 5843  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cen 7896  TopOnctopon 20618   Cn ccn 20938  Homeochmeo 21466  chmph 21467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-map 7804  df-en 7900  df-top 20621  df-topon 20623  df-cn 20941  df-hmeo 21468  df-hmph 21469
This theorem is referenced by: (None)
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