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Theorem indishmph 22382
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 8496 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 6591 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 f1odm 6595 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝐵 → dom 𝑓 = 𝐴)
4 vex 3476 . . . . . . . . . . 11 𝑓 ∈ V
54dmex 7594 . . . . . . . . . 10 dom 𝑓 ∈ V
63, 5eqeltrrdi 2920 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝐴 ∈ V)
7 f1ofo 6598 . . . . . . . . 9 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴onto𝐵)
8 fornex 7635 . . . . . . . . 9 (𝐴 ∈ V → (𝑓:𝐴onto𝐵𝐵 ∈ V))
96, 7, 8sylc 65 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝐵 ∈ V)
109, 6elmapd 8398 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵))
112, 10mpbird 259 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐵m 𝐴))
12 indistopon 21585 . . . . . . . 8 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
136, 12syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ∈ (TopOn‘𝐴))
14 cnindis 21876 . . . . . . 7 (({∅, 𝐴} ∈ (TopOn‘𝐴) ∧ 𝐵 ∈ V) → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵m 𝐴))
1513, 9, 14syl2anc 586 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐴} Cn {∅, 𝐵}) = (𝐵m 𝐴))
1611, 15eleqtrrd 2914 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}))
17 f1ocnv 6603 . . . . . . . 8 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵1-1-onto𝐴)
18 f1of 6591 . . . . . . . 8 (𝑓:𝐵1-1-onto𝐴𝑓:𝐵𝐴)
1917, 18syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐵𝐴)
206, 9elmapd 8398 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
2119, 20mpbird 259 . . . . . 6 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ (𝐴m 𝐵))
22 indistopon 21585 . . . . . . . 8 (𝐵 ∈ V → {∅, 𝐵} ∈ (TopOn‘𝐵))
239, 22syl 17 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐵} ∈ (TopOn‘𝐵))
24 cnindis 21876 . . . . . . 7 (({∅, 𝐵} ∈ (TopOn‘𝐵) ∧ 𝐴 ∈ V) → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴m 𝐵))
2523, 6, 24syl2anc 586 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → ({∅, 𝐵} Cn {∅, 𝐴}) = (𝐴m 𝐵))
2621, 25eleqtrrd 2914 . . . . 5 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴}))
27 ishmeo 22343 . . . . 5 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) ↔ (𝑓 ∈ ({∅, 𝐴} Cn {∅, 𝐵}) ∧ 𝑓 ∈ ({∅, 𝐵} Cn {∅, 𝐴})))
2816, 26, 27sylanbrc 585 . . . 4 (𝑓:𝐴1-1-onto𝐵𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}))
29 hmphi 22361 . . . 4 (𝑓 ∈ ({∅, 𝐴}Homeo{∅, 𝐵}) → {∅, 𝐴} ≃ {∅, 𝐵})
3028, 29syl 17 . . 3 (𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
3130exlimiv 1931 . 2 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
321, 31sylbi 219 1 (𝐴𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1780  wcel 2114  Vcvv 3473  c0 4269  {cpr 4545   class class class wbr 5042  ccnv 5530  dom cdm 5531  wf 6327  ontowfo 6329  1-1-ontowf1o 6330  cfv 6331  (class class class)co 7133  m cmap 8384  cen 8484  TopOnctopon 21494   Cn ccn 21808  Homeochmeo 22337  chmph 22338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-1o 8080  df-map 8386  df-en 8488  df-top 21478  df-topon 21495  df-cn 21811  df-hmeo 22339  df-hmph 22340
This theorem is referenced by: (None)
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