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Theorem hmpher 17475
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 17447 . . . . . 6  |-  ~=  =  ( `'  Homeo  " ( _V  \  1o ) )
2 cnvimass 5033 . . . . . . 7  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 17448 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5343 . . . . . . . 8  |-  (  Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 8 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3210 . . . . . 6  |-  ( `' 
Homeo  " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3208 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 4794 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 4775 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 17 . . . 4  |-  Rel  ~=
1110a1i 10 . . 3  |-  (  T. 
->  Rel  ~=  )
12 hmphsym 17473 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 452 . . 3  |-  ( (  T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 17474 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 452 . . 3  |-  ( (  T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 17472 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 17470 . . . . 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 6686 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   Rel wrel 4694    Fn wfn 5250   1oc1o 6472    Er wer 6657   Topctop 16631    Homeo chmeo 17444    ~= chmph 17445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-er 6660  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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