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Theorem indishmph 22010
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 8250 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 6391 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 f1odm 6395 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
4 vex 3401 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7378 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5syl6eqelr 2868 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
7 f1ofo 6398 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
8 fornex 7414 . . . . . . . . 9 (𝐴 ∈ V → (𝑓:𝐴onto𝐵𝐵 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
109, 6elmapd 8154 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
112, 10mpbird 249 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐵𝑚 𝐴))
12 indistopon 21213 . . . . . . . 8 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14 cnindis 21504 . . . . . . 7 (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1513, 9, 14syl2anc 579 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1611, 15eleqtrrd 2862 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17 f1ocnv 6403 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
18 f1of 6391 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵𝐴)
206, 9elmapd 8154 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
2119, 20mpbird 249 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐴𝑚 𝐵))
22 indistopon 21213 . . . . . . . 8 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24 cnindis 21504 . . . . . . 7 (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2523, 6, 24syl2anc 579 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2621, 25eleqtrrd 2862 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27 ishmeo 21971 . . . . 5 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ 𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
2816, 26, 27sylanbrc 578 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29 hmphi 21989 . . . 4 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
3028, 29syl 17 . . 3 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
3130exlimiv 1973 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
321, 31sylbi 209 1 (𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wex 1823  wcel 2107  Vcvv 3398  c0 4141  {cpr 4400   class class class wbr 4886  ccnv 5354  dom cdm 5355  wf 6131  ontowfo 6133  1-1-ontowf1o 6134  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  cen 8238  TopOnctopon 21122   Cn ccn 21436  Homeochmeo 21965  chmph 21966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-1o 7843  df-map 8142  df-en 8242  df-top 21106  df-topon 21123  df-cn 21439  df-hmeo 21967  df-hmph 21968
This theorem is referenced by: (None)
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