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Theorem dominc 25280
Description: The domain of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
Hypothesis
Ref Expression
dominc.1  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
Assertion
Ref Expression
dominc  |-  dom  C  =  _V
Distinct variable groups:    x, C    x, y
Allowed substitution hint:    C( y)

Proof of Theorem dominc
StepHypRef Expression
1 eqv 3470 . 2  |-  ( dom 
C  =  _V  <->  A. x  x  e.  dom  C )
2 ssid 3197 . . . . 5  |-  x  C_  x
3 sseq2 3200 . . . . . . . 8  |-  ( x  =  y  ->  (
x  C_  x  <->  x  C_  y
) )
43equcoms 1651 . . . . . . 7  |-  ( y  =  x  ->  (
x  C_  x  <->  x  C_  y
) )
54biimpd 198 . . . . . 6  |-  ( y  =  x  ->  (
x  C_  x  ->  x 
C_  y ) )
65spimev 1939 . . . . 5  |-  ( x 
C_  x  ->  E. y  x  C_  y )
72, 6ax-mp 8 . . . 4  |-  E. y  x  C_  y
8 abid 2271 . . . 4  |-  ( x  e.  { x  |  E. y  x  C_  y }  <->  E. y  x  C_  y )
97, 8mpbir 200 . . 3  |-  x  e. 
{ x  |  E. y  x  C_  y }
10 dominc.1 . . . . 5  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
1110dmeqi 4880 . . . 4  |-  dom  C  =  dom  { <. x ,  y >.  |  x 
C_  y }
12 dmopab 4889 . . . 4  |-  dom  { <. x ,  y >.  |  x  C_  y }  =  { x  |  E. y  x  C_  y }
1311, 12eqtri 2303 . . 3  |-  dom  C  =  { x  |  E. y  x  C_  y }
149, 13eleqtrri 2356 . 2  |-  x  e. 
dom  C
151, 14mpgbir 1537 1  |-  dom  C  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   {copab 4076   dom cdm 4689
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-dm 4699
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