MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmpher Structured version   Unicode version

Theorem hmpher 20015
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 19987 . . . . . 6  |-  ~=  =  ( `' Homeo " ( _V  \  1o ) )
2 cnvimass 5350 . . . . . . 7  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 19988 . . . . . . . 8  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5673 . . . . . . . 8  |-  ( Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 5 . . . . . . 7  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3531 . . . . . 6  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3529 . . . . 5  |-  ~=  C_  ( Top  X.  Top )
8 relxp 5103 . . . . 5  |-  Rel  ( Top  X.  Top )
9 relss 5083 . . . . 5  |-  (  ~=  C_  ( Top  X.  Top )  ->  ( Rel  ( Top  X.  Top )  ->  Rel  ~=  ) )
107, 8, 9mp2 9 . . . 4  |-  Rel  ~=
1110a1i 11 . . 3  |-  ( T. 
->  Rel  ~=  )
12 hmphsym 20013 . . . 4  |-  ( x  ~=  y  ->  y  ~=  x )
1312adantl 466 . . 3  |-  ( ( T.  /\  x  ~=  y )  ->  y  ~=  x )
14 hmphtr 20014 . . . 4  |-  ( ( x  ~=  y  /\  y  ~=  z )  ->  x  ~=  z )
1514adantl 466 . . 3  |-  ( ( T.  /\  ( x  ~=  y  /\  y  ~=  z ) )  ->  x  ~=  z )
16 hmphref 20012 . . . . 5  |-  ( x  e.  Top  ->  x  ~=  x )
17 hmphtop1 20010 . . . . 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 7329 . 2  |-  ( T. 
->  ~=  Er  Top )
2120trud 1383 1  |-  ~=  Er  Top
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   T. wtru 1375    e. wcel 1762   _Vcvv 3108    \ cdif 3468    C_ wss 3471   class class class wbr 4442    X. cxp 4992   `'ccnv 4993   dom cdm 4994   "cima 4997   Rel wrel 4999    Fn wfn 5576   1oc1o 7115    Er wer 7300   Topctop 19156   Homeochmeo 19984    ~= chmph 19985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-1o 7122  df-er 7303  df-map 7414  df-top 19161  df-topon 19164  df-cn 19489  df-hmeo 19986  df-hmph 19987
This theorem is referenced by:  ismntop  27632
  Copyright terms: Public domain W3C validator