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Theorem hmphen 23793
Description: Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
hmphen (𝐽𝐾𝐽𝐾)

Proof of Theorem hmphen
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 23784 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4353 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 23768 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4 cntop1 23248 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
53, 4syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top)
6 cntop2 23249 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
73, 6syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top)
8 eqid 2737 . . . . . 6 (𝑥𝐽 ↦ (𝑓𝑥)) = (𝑥𝐽 ↦ (𝑓𝑥))
98hmeoimaf1o 23778 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾)
10 f1oen2g 9009 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾) → 𝐽𝐾)
115, 7, 9, 10syl3anc 1373 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
1211exlimiv 1930 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
132, 12sylbi 217 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝐽𝐾)
141, 13sylbi 217 1 (𝐽𝐾𝐽𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1779  wcel 2108  wne 2940  c0 4333   class class class wbr 5143  cmpt 5225  cima 5688  1-1-ontowf1o 6560  (class class class)co 7431  cen 8982  Topctop 22899   Cn ccn 23232  Homeochmeo 23761  chmph 23762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-en 8986  df-top 22900  df-topon 22917  df-cn 23235  df-hmeo 23763  df-hmph 23764
This theorem is referenced by:  hmph0  23803  hmphindis  23805
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