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Theorem indishmph 23309
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 8951 . 2 (𝐴 β‰ˆ 𝐡 ↔ βˆƒπ‘“ 𝑓:𝐴–1-1-onto→𝐡)
2 f1of 6833 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓:𝐴⟢𝐡)
3 f1odm 6837 . . . . . . . . . 10 (𝑓:𝐴–1-1-onto→𝐡 β†’ dom 𝑓 = 𝐴)
4 vex 3478 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7904 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5eqeltrrdi 2842 . . . . . . . . 9 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝐴 ∈ V)
7 f1ofo 6840 . . . . . . . . 9 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓:𝐴–onto→𝐡)
8 focdmex 7944 . . . . . . . . 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 8836 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ (𝑓 ∈ (𝐡 ↑m 𝐴) ↔ 𝑓:𝐴⟢𝐡))
112, 10mpbird 256 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ (𝐡 ↑m 𝐴))
12 indistopon 22511 . . . . . . . 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 22803 . . . . . . 7 (({βˆ…, 𝐴} ∈ (TopOnβ€˜π΄) ∧ 𝐡 ∈ V) β†’ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) = (𝐡 ↑m 𝐴))
1513, 9, 14syl2anc 584 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) = (𝐡 ↑m 𝐴))
1611, 15eleqtrrd 2836 . . . . 5 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}))
17 f1ocnv 6845 . . . . . . . 8 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓:𝐡–1-1-onto→𝐴)
18 f1of 6833 . . . . . . . 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 8836 . . . . . . 7 (𝑓:𝐴–1-1-onto→𝐡 β†’ (◑𝑓 ∈ (𝐴 ↑m 𝐡) ↔ ◑𝑓:𝐡⟢𝐴))
2119, 20mpbird 256 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓 ∈ (𝐴 ↑m 𝐡))
22 indistopon 22511 . . . . . . . 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 22803 . . . . . . 7 (({βˆ…, 𝐡} ∈ (TopOnβ€˜π΅) ∧ 𝐴 ∈ V) β†’ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}) = (𝐴 ↑m 𝐡))
2523, 6, 24syl2anc 584 . . . . . 6 (𝑓:𝐴–1-1-onto→𝐡 β†’ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}) = (𝐴 ↑m 𝐡))
2621, 25eleqtrrd 2836 . . . . 5 (𝑓:𝐴–1-1-onto→𝐡 β†’ ◑𝑓 ∈ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴}))
27 ishmeo 23270 . . . . 5 (𝑓 ∈ ({βˆ…, 𝐴}Homeo{βˆ…, 𝐡}) ↔ (𝑓 ∈ ({βˆ…, 𝐴} Cn {βˆ…, 𝐡}) ∧ ◑𝑓 ∈ ({βˆ…, 𝐡} Cn {βˆ…, 𝐴})))
2816, 26, 27sylanbrc 583 . . . 4 (𝑓:𝐴–1-1-onto→𝐡 β†’ 𝑓 ∈ ({βˆ…, 𝐴}Homeo{βˆ…, 𝐡}))
29 hmphi 23288 . . . 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 1933 . 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 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474  βˆ…c0 4322  {cpr 4630   class class class wbr 5148  β—‘ccnv 5675  dom cdm 5676  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822   β‰ˆ cen 8938  TopOnctopon 22419   Cn ccn 22735  Homeochmeo 23264   ≃ chmph 23265
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-1o 8468  df-map 8824  df-en 8942  df-top 22403  df-topon 22420  df-cn 22738  df-hmeo 23266  df-hmph 23267
This theorem is referenced by: (None)
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