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Theorem indishmph 23746
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 8974 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 6838 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 f1odm 6842 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
4 vex 3465 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7917 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5eqeltrrdi 2834 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
7 f1ofo 6845 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
8 focdmex 7960 . . . . . . . . 9 (𝐴 ∈ V → (𝑓:𝐴onto𝐵𝐵 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
109, 6elmapd 8859 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
112, 10mpbird 256 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐵m 𝐴))
12 indistopon 22948 . . . . . . . 8 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14 cnindis 23240 . . . . . . 7 (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵m 𝐴))
1513, 9, 14syl2anc 582 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵m 𝐴))
1611, 15eleqtrrd 2828 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17 f1ocnv 6850 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
18 f1of 6838 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵𝐴)
206, 9elmapd 8859 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
2119, 20mpbird 256 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐴m 𝐵))
22 indistopon 22948 . . . . . . . 8 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24 cnindis 23240 . . . . . . 7 (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴m 𝐵))
2523, 6, 24syl2anc 582 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴m 𝐵))
2621, 25eleqtrrd 2828 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27 ishmeo 23707 . . . . 5 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ 𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
2816, 26, 27sylanbrc 581 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29 hmphi 23725 . . . 4 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
3028, 29syl 17 . . 3 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
3130exlimiv 1925 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
321, 31sylbi 216 1 (𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  c0 4322  {cpr 4632   class class class wbr 5149  ccnv 5677  dom cdm 5678  wf 6545  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  m cmap 8845  cen 8961  TopOnctopon 22856   Cn ccn 23172  Homeochmeo 23701  chmph 23702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-1o 8487  df-map 8847  df-en 8965  df-top 22840  df-topon 22857  df-cn 23175  df-hmeo 23703  df-hmph 23704
This theorem is referenced by: (None)
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