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Theorem hmphtop 20404
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmphtop  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )

Proof of Theorem hmphtop
StepHypRef Expression
1 df-hmph 20382 . . 3  |-  ~=  =  ( `' Homeo " ( _V  \  1o ) )
2 cnvimass 5367 . . . 4  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  dom  Homeo
3 hmeofn 20383 . . . . 5  |-  Homeo  Fn  ( Top  X.  Top )
4 fndm 5686 . . . . 5  |-  ( Homeo  Fn  ( Top  X.  Top )  ->  dom  Homeo  =  ( Top  X.  Top )
)
53, 4ax-mp 5 . . . 4  |-  dom  Homeo  =  ( Top  X.  Top )
62, 5sseqtri 3531 . . 3  |-  ( `'
Homeo " ( _V  \  1o ) )  C_  ( Top  X.  Top )
71, 6eqsstri 3529 . 2  |-  ~=  C_  ( Top  X.  Top )
87brel 5057 1  |-  ( J  ~=  K  ->  ( J  e.  Top  /\  K  e.  Top ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468   class class class wbr 4456    X. cxp 5006   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   1oc1o 7141   Topctop 19520   Homeochmeo 20379    ~= chmph 20380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-hmeo 20381  df-hmph 20382
This theorem is referenced by:  hmphtop1  20405  hmphtop2  20406
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