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Theorem hmphtr 17474
Description: "Is homeomorph to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphtr  |-  ( ( J  ~=  K  /\  K  ~=  L )  ->  J  ~=  L )

Proof of Theorem hmphtr
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 17467 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 hmph 17467 . 2  |-  ( K  ~=  L  <->  ( K  Homeo  L )  =/=  (/) )
3 n0 3464 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
4 n0 3464 . . 3  |-  ( ( K  Homeo  L )  =/=  (/)  <->  E. g  g  e.  ( K  Homeo  L ) )
5 eeanv 1854 . . . 4  |-  ( E. f E. g ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  <->  ( E. f 
f  e.  ( J 
Homeo  K )  /\  E. g  g  e.  ( K  Homeo  L ) ) )
6 hmeoco 17463 . . . . . 6  |-  ( ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  ( g  o.  f )  e.  ( J  Homeo  L )
)
7 hmphi 17468 . . . . . 6  |-  ( ( g  o.  f )  e.  ( J  Homeo  L )  ->  J  ~=  L )
86, 7syl 15 . . . . 5  |-  ( ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
98exlimivv 1667 . . . 4  |-  ( E. f E. g ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
105, 9sylbir 204 . . 3  |-  ( ( E. f  f  e.  ( J  Homeo  K )  /\  E. g  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
113, 4, 10syl2anb 465 . 2  |-  ( ( ( J  Homeo  K )  =/=  (/)  /\  ( K 
Homeo  L )  =/=  (/) )  ->  J  ~=  L )
121, 2, 11syl2anb 465 1  |-  ( ( J  ~=  K  /\  K  ~=  L )  ->  J  ~=  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023    o. ccom 4693  (class class class)co 5858    Homeo chmeo 17444    ~= chmph 17445
This theorem is referenced by:  hmpher  17475  xrhmph  18445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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