MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmphtr Unicode version

Theorem hmphtr 17738
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 17731 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 hmph 17731 . 2  |-  ( K  ~=  L  <->  ( K  Homeo  L )  =/=  (/) )
3 n0 3582 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
4 n0 3582 . . 3  |-  ( ( K  Homeo  L )  =/=  (/)  <->  E. g  g  e.  ( K  Homeo  L ) )
5 eeanv 1926 . . . 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 17727 . . . . . 6  |-  ( ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  ( g  o.  f )  e.  ( J  Homeo  L )
)
7 hmphi 17732 . . . . . 6  |-  ( ( g  o.  f )  e.  ( J  Homeo  L )  ->  J  ~=  L )
86, 7syl 16 . . . . 5  |-  ( ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
98exlimivv 1642 . . . 4  |-  ( E. f E. g ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
105, 9sylbir 205 . . 3  |-  ( ( E. f  f  e.  ( J  Homeo  K )  /\  E. g  g  e.  ( K  Homeo  L ) )  ->  J  ~=  L )
113, 4, 10syl2anb 466 . 2  |-  ( ( ( J  Homeo  K )  =/=  (/)  /\  ( K 
Homeo  L )  =/=  (/) )  ->  J  ~=  L )
121, 2, 11syl2anb 466 1  |-  ( ( J  ~=  K  /\  K  ~=  L )  ->  J  ~=  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    e. wcel 1717    =/= wne 2552   (/)c0 3573   class class class wbr 4155    o. ccom 4824  (class class class)co 6022    Homeo chmeo 17708    ~= chmph 17709
This theorem is referenced by:  hmpher  17739  xrhmph  18845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-1o 6662  df-map 6958  df-top 16888  df-topon 16891  df-cn 17215  df-hmeo 17710  df-hmph 17711
  Copyright terms: Public domain W3C validator