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Theorem hmphen2 23747
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 23724 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4346 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 cmphaushmeo.1 . . . . . 6 𝑋 = 𝐽
4 cmphaushmeo.2 . . . . . 6 𝑌 = 𝐾
53, 4hmeof1o 23712 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋1-1-onto𝑌)
6 f1oen3g 8987 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋1-1-onto𝑌) → 𝑋𝑌)
75, 6mpdan 685 . . . 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 2929  c0 4322   cuni 4909   class class class wbr 5149  1-1-ontowf1o 6548  (class class class)co 7419  cen 8961  Homeochmeo 23701  chmph 23702
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-1o 8487  df-map 8847  df-en 8965  df-top 22840  df-topon 22857  df-cn 23175  df-hmeo 23703  df-hmph 23704
This theorem is referenced by: (None)
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