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Theorem hmphen2 22406
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 22383 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4309 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 cmphaushmeo.1 . . . . . 6 𝑋 = 𝐽
4 cmphaushmeo.2 . . . . . 6 𝑌 = 𝐾
53, 4hmeof1o 22371 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋1-1-onto𝑌)
6 f1oen3g 8524 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋1-1-onto𝑌) → 𝑋𝑌)
75, 6mpdan 685 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
87exlimiv 1927 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋𝑌)
92, 8sylbi 219 . 2 ((𝐽Homeo𝐾) ≠ ∅ → 𝑋𝑌)
101, 9sylbi 219 1 (𝐽𝐾𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1776  wcel 2110  wne 3016  c0 4290   cuni 4837   class class class wbr 5065  1-1-ontowf1o 6353  (class class class)co 7155  cen 8505  Homeochmeo 22360  chmph 22361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-1o 8101  df-map 8407  df-en 8509  df-top 21501  df-topon 21518  df-cn 21834  df-hmeo 22362  df-hmph 22363
This theorem is referenced by: (None)
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