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Theorem hmphen 21636
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 21627 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 3964 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 21611 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4 cntop1 21092 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
53, 4syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top)
6 cntop2 21093 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
73, 6syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top)
8 eqid 2651 . . . . . 6 (𝑥𝐽 ↦ (𝑓𝑥)) = (𝑥𝐽 ↦ (𝑓𝑥))
98hmeoimaf1o 21621 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾)
10 f1oen2g 8014 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾) → 𝐽𝐾)
115, 7, 9, 10syl3anc 1366 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
1211exlimiv 1898 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
132, 12sylbi 207 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝐽𝐾)
141, 13sylbi 207 1 (𝐽𝐾𝐽𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1744  wcel 2030  wne 2823  c0 3948   class class class wbr 4685  cmpt 4762  cima 5146  1-1-ontowf1o 5925  (class class class)co 6690  cen 7994  Topctop 20746   Cn ccn 21076  Homeochmeo 21604  chmph 21605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-map 7901  df-en 7998  df-top 20747  df-topon 20764  df-cn 21079  df-hmeo 21606  df-hmph 21607
This theorem is referenced by:  hmph0  21646  hmphindis  21648
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