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Theorem relrelss 5196
Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
relrelss  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )

Proof of Theorem relrelss
StepHypRef Expression
1 df-rel 4696 . . 3  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
21anbi2i 675 . 2  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V )
) )
3 relssdmrn 5193 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
4 ssv 3198 . . . . 5  |-  ran  A  C_ 
_V
5 xpss12 4792 . . . . 5  |-  ( ( dom  A  C_  ( _V  X.  _V )  /\  ran  A  C_  _V )  ->  ( dom  A  X.  ran  A )  C_  (
( _V  X.  _V )  X.  _V ) )
64, 5mpan2 652 . . . 4  |-  ( dom 
A  C_  ( _V  X.  _V )  ->  ( dom  A  X.  ran  A
)  C_  ( ( _V  X.  _V )  X. 
_V ) )
73, 6sylan9ss 3192 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  A  C_  ( ( _V  X.  _V )  X.  _V )
)
8 xpss 4793 . . . . . 6  |-  ( ( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
9 sstr 3187 . . . . . 6  |-  ( ( A  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
)  ->  A  C_  ( _V  X.  _V ) )
108, 9mpan2 652 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  A  C_  ( _V  X.  _V ) )
11 df-rel 4696 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1210, 11sylibr 203 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  Rel  A )
13 dmss 4878 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  dom  ( ( _V  X.  _V )  X.  _V )
)
14 vn0 3462 . . . . . 6  |-  _V  =/=  (/)
15 dmxp 4897 . . . . . 6  |-  ( _V  =/=  (/)  ->  dom  ( ( _V  X.  _V )  X.  _V )  =  ( _V  X.  _V )
)
1614, 15ax-mp 8 . . . . 5  |-  dom  (
( _V  X.  _V )  X.  _V )  =  ( _V  X.  _V )
1713, 16syl6sseq 3224 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  ( _V  X.  _V ) )
1812, 17jca 518 . . 3  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V ) ) )
197, 18impbii 180 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
202, 19bitri 240 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  dftpos3  6252  tpostpos2  6255
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-ral 2548  df-rex 2549  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-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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