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Theorem hmphen 22396
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 22387 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4293 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 22371 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4 cntop1 21851 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
53, 4syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top)
6 cntop2 21852 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
73, 6syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top)
8 eqid 2824 . . . . . 6 (𝑥𝐽 ↦ (𝑓𝑥)) = (𝑥𝐽 ↦ (𝑓𝑥))
98hmeoimaf1o 22381 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾)
10 f1oen2g 8522 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥𝐽 ↦ (𝑓𝑥)):𝐽1-1-onto𝐾) → 𝐽𝐾)
115, 7, 9, 10syl3anc 1368 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
1211exlimiv 1932 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
132, 12sylbi 220 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝐽𝐾)
141, 13sylbi 220 1 (𝐽𝐾𝐽𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1781  wcel 2115  wne 3014  c0 4276   class class class wbr 5052  cmpt 5132  cima 5545  1-1-ontowf1o 6342  (class class class)co 7149  cen 8502  Topctop 21504   Cn ccn 21835  Homeochmeo 22364  chmph 22365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-1o 8098  df-map 8404  df-en 8506  df-top 21505  df-topon 21522  df-cn 21838  df-hmeo 22366  df-hmph 22367
This theorem is referenced by:  hmph0  22406  hmphindis  22408
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