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Theorem hmpher 19356
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 19328 . . . . . 6  |-  ~=  =  ( `' Homeo " ( _V  \  1o ) )
2 cnvimass 5188 . . . . . . 7  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 19329 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5509 . . . . . . . 8  |-  ( Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 5 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3387 . . . . . 6  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3385 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 4946 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 4926 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=  )
12 hmphsym 19354 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 466 . . 3  |-  ( ( T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 19355 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 466 . . 3  |-  ( ( T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 19353 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 19351 . . . . 5  |-  ( x  ~=  x  ->  x  e.  Top )
1816, 17impbii 188 . . . 4  |-  ( x  e.  Top  <->  x  ~=  x )
1918a1i 11 . . 3  |-  ( T. 
->  ( x  e.  Top  <->  x  ~=  x ) )
2011, 13, 15, 19iserd 7126 . 2  |-  ( T. 
->  ~=  Er  Top )
2120trud 1378 1  |-  ~=  Er  Top
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756   _Vcvv 2971    \ cdif 3324    C_ wss 3327   class class class wbr 4291    X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842   Rel wrel 4844    Fn wfn 5412   1oc1o 6912    Er wer 7097   Topctop 18497   Homeochmeo 19325    ~= chmph 19326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-1o 6919  df-er 7100  df-map 7215  df-top 18502  df-topon 18505  df-cn 18830  df-hmeo 19327  df-hmph 19328
This theorem is referenced by: (None)
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