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Theorem hmpher 17491
Description: "Is homeomorph 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 17463 . . . . . 6  |-  ~=  =  ( `'  Homeo  " ( _V  \  1o ) )
2 cnvimass 5049 . . . . . . 7  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 17464 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5359 . . . . . . . 8  |-  (  Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 8 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3223 . . . . . 6  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3221 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 4810 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 4791 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 17 . . . 4  |-  Rel  ~=
1110a1i 10 . . 3  |-  (  T. 
->  Rel  ~=  )
12 hmphsym 17489 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 452 . . 3  |-  ( (  T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 17490 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 452 . . 3  |-  ( (  T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 17488 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 17486 . . . . 5  |-  ( x  ~=  x  ->  x  e.  Top )
1816, 17impbii 180 . . . 4  |-  ( x  e.  Top  <->  x  ~=  x )
1918a1i 10 . . 3  |-  (  T. 
->  ( x  e.  Top  <->  x  ~=  x ) )
2011, 13, 15, 19iserd 6702 . 2  |-  (  T. 
->  ~=  Er  Top )
2120trud 1314 1  |-  ~=  Er  Top
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708   Rel wrel 4710    Fn wfn 5266   1oc1o 6488    Er wer 6673   Topctop 16647    Homeo chmeo 17460    ~= chmph 17461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-er 6676  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462  df-hmph 17463
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