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Theorem relrelss 5347
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 4830 . . 3  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
21anbi2i 692 . 2  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V )
) )
3 relssdmrn 5344 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
4 ssv 3462 . . . . 5  |-  ran  A  C_ 
_V
5 xpss12 4929 . . . . 5  |-  ( ( dom  A  C_  ( _V  X.  _V )  /\  ran  A  C_  _V )  ->  ( dom  A  X.  ran  A )  C_  (
( _V  X.  _V )  X.  _V ) )
64, 5mpan2 669 . . . 4  |-  ( dom 
A  C_  ( _V  X.  _V )  ->  ( dom  A  X.  ran  A
)  C_  ( ( _V  X.  _V )  X. 
_V ) )
73, 6sylan9ss 3455 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  A  C_  ( ( _V  X.  _V )  X.  _V )
)
8 xpss 4930 . . . . . 6  |-  ( ( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
9 sstr 3450 . . . . . 6  |-  ( ( A  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
)  ->  A  C_  ( _V  X.  _V ) )
108, 9mpan2 669 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  A  C_  ( _V  X.  _V ) )
11 df-rel 4830 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1210, 11sylibr 212 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  Rel  A )
13 dmss 5023 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  dom  ( ( _V  X.  _V )  X.  _V )
)
14 vn0 3746 . . . . . 6  |-  _V  =/=  (/)
15 dmxp 5042 . . . . . 6  |-  ( _V  =/=  (/)  ->  dom  ( ( _V  X.  _V )  X.  _V )  =  ( _V  X.  _V )
)
1614, 15ax-mp 5 . . . . 5  |-  dom  (
( _V  X.  _V )  X.  _V )  =  ( _V  X.  _V )
1713, 16syl6sseq 3488 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  ( _V  X.  _V ) )
1812, 17jca 530 . . 3  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V ) ) )
197, 18impbii 187 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
202, 19bitri 249 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    =/= wne 2598   _Vcvv 3059    C_ wss 3414   (/)c0 3738    X. cxp 4821   dom cdm 4823   ran crn 4824   Rel wrel 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834
This theorem is referenced by:  dftpos3  6976  tpostpos2  6979
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