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Theorem indishmph 21803
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 8130 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 6298 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 f1odm 6302 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
4 vex 3343 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7264 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5syl6eqelr 2848 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
7 f1ofo 6305 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
8 fornex 7300 . . . . . . . . 9 (𝐴 ∈ V → (𝑓:𝐴onto𝐵𝐵 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
109, 6elmapd 8037 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
112, 10mpbird 247 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐵𝑚 𝐴))
12 indistopon 21007 . . . . . . . 8 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14 cnindis 21298 . . . . . . 7 (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1513, 9, 14syl2anc 696 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵𝑚 𝐴))
1611, 15eleqtrrd 2842 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17 f1ocnv 6310 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
18 f1of 6298 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵𝐴)
206, 9elmapd 8037 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
2119, 20mpbird 247 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐴𝑚 𝐵))
22 indistopon 21007 . . . . . . . 8 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24 cnindis 21298 . . . . . . 7 (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2523, 6, 24syl2anc 696 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴𝑚 𝐵))
2621, 25eleqtrrd 2842 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27 ishmeo 21764 . . . . 5 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ 𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
2816, 26, 27sylanbrc 701 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29 hmphi 21782 . . . 4 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
3028, 29syl 17 . . 3 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
3130exlimiv 2007 . 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 1632  wex 1853  wcel 2139  Vcvv 3340  c0 4058  {cpr 4323   class class class wbr 4804  ccnv 5265  dom cdm 5266  wf 6045  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  cen 8118  TopOnctopon 20917   Cn ccn 21230  Homeochmeo 21758  chmph 21759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-1o 7729  df-map 8025  df-en 8122  df-top 20901  df-topon 20918  df-cn 21233  df-hmeo 21760  df-hmph 21761
This theorem is referenced by: (None)
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