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Theorem hmpher 20577
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 20549 . . . . . 6  |-  ~=  =  ( `' Homeo " ( _V  \  1o ) )
2 cnvimass 5177 . . . . . . 7  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 20550 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5661 . . . . . . . 8  |-  ( Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 5 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3474 . . . . . 6  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3472 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 4931 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 4911 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=  )
12 hmphsym 20575 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 464 . . 3  |-  ( ( T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 20576 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 464 . . 3  |-  ( ( T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 20574 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 20572 . . . . 5  |-  ( x  ~=  x  ->  x  e.  Top )
1816, 17impbii 187 . . . 4  |-  ( x  e.  Top  <->  x  ~=  x )
1918a1i 11 . . 3  |-  ( T. 
->  ( x  e.  Top  <->  x  ~=  x ) )
2011, 13, 15, 19iserd 7374 . 2  |-  ( T. 
->  ~=  Er  Top )
2120trud 1414 1  |-  ~=  Er  Top
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842   _Vcvv 3059    \ cdif 3411    C_ wss 3414   class class class wbr 4395    X. cxp 4821   `'ccnv 4822   dom cdm 4823   "cima 4826   Rel wrel 4828    Fn wfn 5564   1oc1o 7160    Er wer 7345   Topctop 19686   Homeochmeo 20546    ~= chmph 20547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-1o 7167  df-er 7348  df-map 7459  df-top 19691  df-topon 19694  df-cn 20021  df-hmeo 20548  df-hmph 20549
This theorem is referenced by:  ismntop  28456
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