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Theorem relrelss 4872
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 4380 . . 3  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
21anbi2i 438 . 2  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V )
) )
3 relssdmrn 4869 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
4 ssv 2993 . . . . 5  |-  ran  A  C_ 
_V
5 xpss12 4473 . . . . 5  |-  ( ( dom  A  C_  ( _V  X.  _V )  /\  ran  A  C_  _V )  ->  ( dom  A  X.  ran  A )  C_  (
( _V  X.  _V )  X.  _V ) )
64, 5mpan2 409 . . . 4  |-  ( dom 
A  C_  ( _V  X.  _V )  ->  ( dom  A  X.  ran  A
)  C_  ( ( _V  X.  _V )  X. 
_V ) )
73, 6sylan9ss 2986 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  A  C_  ( ( _V  X.  _V )  X.  _V )
)
8 xpss 4474 . . . . . 6  |-  ( ( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
9 sstr 2981 . . . . . 6  |-  ( ( A  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
)  ->  A  C_  ( _V  X.  _V ) )
108, 9mpan2 409 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  A  C_  ( _V  X.  _V ) )
11 df-rel 4380 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1210, 11sylibr 141 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  Rel  A )
13 dmss 4562 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  dom  ( ( _V  X.  _V )  X.  _V )
)
14 vn0m 3260 . . . . . 6  |-  E. x  x  e.  _V
15 dmxpm 4583 . . . . . 6  |-  ( E. x  x  e.  _V  ->  dom  ( ( _V 
X.  _V )  X.  _V )  =  ( _V  X.  _V ) )
1614, 15ax-mp 7 . . . . 5  |-  dom  (
( _V  X.  _V )  X.  _V )  =  ( _V  X.  _V )
1713, 16syl6sseq 3019 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  ( _V  X.  _V ) )
1812, 17jca 294 . . 3  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V ) ) )
197, 18impbii 121 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
202, 19bitri 177 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574    C_ wss 2945    X. cxp 4371   dom cdm 4373   ran crn 4374   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  dftpos3  5908  tpostpos2  5911
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