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Theorem indishmph 23165
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 (𝐴 β‰ˆ 𝐡 β†’ {βˆ…, 𝐴} ≃ {βˆ…, 𝐡})

Proof of Theorem indishmph
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8896 . 2 (𝐴 β‰ˆ 𝐡 ↔ βˆƒπ‘“ 𝑓:𝐴–1-1-onto→𝐡)
2 f1of 6785 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓:𝐴⟢𝐡)
3 f1odm 6789 . . . . . . . . . 10 (𝑓:𝐴–1-1-onto→𝐡 β†’ dom 𝑓 = 𝐴)
4 vex 3448 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7849 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5eqeltrrdi 2843 . . . . . . . . 9 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝐴 ∈ V)
7 f1ofo 6792 . . . . . . . . 9 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓:𝐴–onto→𝐡)
8 focdmex 7889 . . . . . . . . 9 (𝐴 ∈ V β†’ (𝑓:𝐴–onto→𝐡 β†’ 𝐡 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝐡 ∈ V)
109, 6elmapd 8782 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ (𝑓 ∈ (𝐡 ↑m 𝐴) ↔ 𝑓:𝐴⟢𝐡))
112, 10mpbird 257 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ (𝐡 ↑m 𝐴))
12 indistopon 22367 . . . . . . . 8 (𝐴 ∈ V β†’ {βˆ…, 𝐴} ∈ (TopOnβ€˜π΄))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ {βˆ…, 𝐴} ∈ (TopOnβ€˜π΄))
14 cnindis 22659 . . . . . . 7 (({βˆ…, 𝐴} ∈ (TopOnβ€˜π΄) ∧ 𝐡 ∈ V) β†’ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) = (𝐡 ↑m 𝐴))
1513, 9, 14syl2anc 585 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) = (𝐡 ↑m 𝐴))
1611, 15eleqtrrd 2837 . . . . 5 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}))
17 f1ocnv 6797 . . . . . . . 8 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓:𝐡–1-1-onto→𝐴)
18 f1of 6785 . . . . . . . 8 (◑𝑓:𝐡–1-1-onto→𝐴 β†’ ◑𝑓:𝐡⟢𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓:𝐡⟢𝐴)
206, 9elmapd 8782 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ (◑𝑓 ∈ (𝐴 ↑m 𝐡) ↔ ◑𝑓:𝐡⟢𝐴))
2119, 20mpbird 257 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓 ∈ (𝐴 ↑m 𝐡))
22 indistopon 22367 . . . . . . . 8 (𝐡 ∈ V β†’ {βˆ…, 𝐡} ∈ (TopOnβ€˜π΅))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ {βˆ…, 𝐡} ∈ (TopOnβ€˜π΅))
24 cnindis 22659 . . . . . . 7 (({βˆ…, 𝐡} ∈ (TopOnβ€˜π΅) ∧ 𝐴 ∈ V) β†’ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}) = (𝐴 ↑m 𝐡))
2523, 6, 24syl2anc 585 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}) = (𝐴 ↑m 𝐡))
2621, 25eleqtrrd 2837 . . . . 5 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓 ∈ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}))
27 ishmeo 23126 . . . . 5 (𝑓 ∈ ({βˆ…, 𝐴}Homeo{βˆ…, 𝐡}) ↔ (𝑓 ∈ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) ∧ ◑𝑓 ∈ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴})))
2816, 26, 27sylanbrc 584 . . . 4 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ ({βˆ…, 𝐴}Homeo{βˆ…, 𝐡}))
29 hmphi 23144 . . . 4 (𝑓 ∈ ({βˆ…, 𝐴}Homeo{βˆ…, 𝐡}) β†’ {βˆ…, 𝐴} ≃ {βˆ…, 𝐡})
3028, 29syl 17 . . 3 (𝑓:𝐴–1-1-onto→𝐡 β†’ {βˆ…, 𝐴} ≃ {βˆ…, 𝐡})
3130exlimiv 1934 . 2 (βˆƒπ‘“ 𝑓:𝐴–1-1-onto→𝐡 β†’ {βˆ…, 𝐴} ≃ {βˆ…, 𝐡})
321, 31sylbi 216 1 (𝐴 β‰ˆ 𝐡 β†’ {βˆ…, 𝐴} ≃ {βˆ…, 𝐡})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3444  βˆ…c0 4283  {cpr 4589   class class class wbr 5106  β—‘ccnv 5633  dom cdm 5634  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768   β‰ˆ cen 8883  TopOnctopon 22275   Cn ccn 22591  Homeochmeo 23120   ≃ chmph 23121
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-1o 8413  df-map 8770  df-en 8887  df-top 22259  df-topon 22276  df-cn 22594  df-hmeo 23122  df-hmph 23123
This theorem is referenced by: (None)
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