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Theorem hmphtr 17807
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 17800 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 hmph 17800 . 2  |-  ( K  ~=  L  <->  ( K  Homeo  L )  =/=  (/) )
3 n0 3629 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
4 n0 3629 . . 3  |-  ( ( K  Homeo  L )  =/=  (/)  <->  E. g  g  e.  ( K  Homeo  L ) )
5 eeanv 1937 . . . 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 17796 . . . . . 6  |-  ( ( f  e.  ( J 
Homeo  K )  /\  g  e.  ( K  Homeo  L ) )  ->  ( g  o.  f )  e.  ( J  Homeo  L )
)
7 hmphi 17801 . . . . . 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 1645 . . . 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 1550    e. wcel 1725    =/= wne 2598   (/)c0 3620   class class class wbr 4204    o. ccom 4874  (class class class)co 6073    Homeo chmeo 17777    ~= chmph 17778
This theorem is referenced by:  hmpher  17808  xrhmph  18964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-map 7012  df-top 16955  df-topon 16958  df-cn 17283  df-hmeo 17779  df-hmph 17780
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