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Theorem hmphen2 21650
 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 21627 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 3964 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 cmphaushmeo.1 . . . . . 6 𝑋 = 𝐽
4 cmphaushmeo.2 . . . . . 6 𝑌 = 𝐾
53, 4hmeof1o 21615 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋1-1-onto𝑌)
6 f1oen3g 8013 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋1-1-onto𝑌) → 𝑋𝑌)
75, 6mpdan 703 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
87exlimiv 1898 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
92, 8sylbi 207 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝑋𝑌)
101, 9sylbi 207 1 (𝐽𝐾𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  ∪ cuni 4468   class class class wbr 4685  –1-1-onto→wf1o 5925  (class class class)co 6690   ≈ cen 7994  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: (None)
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