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Theorem dmhmph 25533
Description:  ~= is a relation whose domain is included in  Top. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
dmhmph  |-  dom  ~=  C_ 
Top

Proof of Theorem dmhmph
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  x  e. 
_V
21eldm 4876 . . 3  |-  ( x  e.  dom  ~=  <->  E. y  x  ~=  y )
3 hmphtop1 17470 . . . 4  |-  ( x  ~=  y  ->  x  e.  Top )
43exlimiv 1666 . . 3  |-  ( E. y  x  ~=  y  ->  x  e.  Top )
52, 4sylbi 187 . 2  |-  ( x  e.  dom  ~=  ->  x  e.  Top )
65ssriv 3184 1  |-  dom  ~=  C_ 
Top
Colors of variables: wff set class
Syntax hints:   E.wex 1528    e. wcel 1684    C_ wss 3152   class class class wbr 4023   dom cdm 4689   Topctop 16631    ~= 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-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-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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-hmeo 17446  df-hmph 17447
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