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Theorem hmphen2 23658
Description: Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
cmphaushmeo.1 𝑋 = 𝐽
cmphaushmeo.2 𝑌 = 𝐾
Assertion
Ref Expression
hmphen2 (𝐽𝐾𝑋𝑌)

Proof of Theorem hmphen2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 hmph 23635 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4341 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 cmphaushmeo.1 . . . . . 6 𝑋 = 𝐽
4 cmphaushmeo.2 . . . . . 6 𝑌 = 𝐾
53, 4hmeof1o 23623 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋1-1-onto𝑌)
6 f1oen3g 8964 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋1-1-onto𝑌) → 𝑋𝑌)
75, 6mpdan 684 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
87exlimiv 1925 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
92, 8sylbi 216 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝑋𝑌)
101, 9sylbi 216 1 (𝐽𝐾𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1773  wcel 2098  wne 2934  c0 4317   cuni 4902   class class class wbr 5141  1-1-ontowf1o 6536  (class class class)co 7405  cen 8938  Homeochmeo 23612  chmph 23613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-1o 8467  df-map 8824  df-en 8942  df-top 22751  df-topon 22768  df-cn 23086  df-hmeo 23614  df-hmph 23615
This theorem is referenced by: (None)
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