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Theorem hmpher 20791
Description: "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmpher  |-  ~=  Er  Top

Proof of Theorem hmpher
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hmph 20763 . . . . . 6  |-  ~=  =  ( `' Homeo " ( _V  \  1o ) )
2 cnvimass 5205 . . . . . . 7  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 20764 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5691 . . . . . . . 8  |-  ( Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 5 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3497 . . . . . 6  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3495 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 4959 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 4939 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=  )
12 hmphsym 20789 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 468 . . 3  |-  ( ( T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 20790 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 468 . . 3  |-  ( ( T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 20788 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 20786 . . . . 5  |-  ( x  ~=  x  ->  x  e.  Top )
1816, 17impbii 191 . . . 4  |-  ( x  e.  Top  <->  x  ~=  x )
1918a1i 11 . . 3  |-  ( T. 
->  ( x  e.  Top  <->  x  ~=  x ) )
2011, 13, 15, 19iserd 7395 . 2  |-  ( T. 
->  ~=  Er  Top )
2120trud 1447 1  |-  ~=  Er  Top
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438   T. wtru 1439    e. wcel 1869   _Vcvv 3082    \ cdif 3434    C_ wss 3437   class class class wbr 4421    X. cxp 4849   `'ccnv 4850   dom cdm 4851   "cima 4854   Rel wrel 4856    Fn wfn 5594   1oc1o 7181    Er wer 7366   Topctop 19909   Homeochmeo 20760    ~= chmph 20761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-1o 7188  df-er 7369  df-map 7480  df-top 19913  df-topon 19915  df-cn 20235  df-hmeo 20762  df-hmph 20763
This theorem is referenced by:  ismntop  28832
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