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Theorem dmhmph 24932
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem dmhmph
StepHypRef Expression
1 vex 2792 . . . 4  |-  x  e. 
_V
21eldm 4875 . . 3  |-  ( x  e.  dom  ~=  <->  E. y  x  ~=  y )
3 hmphtop1 17464 . . . 4  |-  ( x  ~=  y  ->  x  e.  Top )
43exlimiv 1667 . . 3  |-  ( E. y  x  ~=  y  ->  x  e.  Top )
52, 4sylbi 189 . 2  |-  ( x  e.  dom  ~=  ->  x  e.  Top )
65ssriv 3185 1  |-  dom  ~=  C_ 
Top
Colors of variables: wff set class
Syntax hints:   E.wex 1529    e. wcel 1685    C_ wss 3153   class class class wbr 4024   dom cdm 4688   Topctop 16625    ~= chmph 17439
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-hmeo 17440  df-hmph 17441
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