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Theorem hmphi 22926
Description: If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphi (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)

Proof of Theorem hmphi
StepHypRef Expression
1 ne0i 4274 . 2 (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽Homeo𝐾) ≠ ∅)
2 hmph 22925 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
31, 2sylibr 233 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wne 2945  c0 4262   class class class wbr 5079  (class class class)co 7271  Homeochmeo 22902  chmph 22903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-1o 8288  df-hmeo 22904  df-hmph 22905
This theorem is referenced by:  hmphref  22930  hmphsym  22931  hmphtr  22932  indishmph  22947  ptcmpfi  22962  t0kq  22967  kqhmph  22968  xrhmph  24108  xrge0hmph  31878  reheibor  35993
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