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Theorem relrelss 5394
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 4886 . . 3  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
21anbi2i 677 . 2  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V )
) )
3 relssdmrn 5391 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
4 ssv 3369 . . . . 5  |-  ran  A  C_ 
_V
5 xpss12 4982 . . . . 5  |-  ( ( dom  A  C_  ( _V  X.  _V )  /\  ran  A  C_  _V )  ->  ( dom  A  X.  ran  A )  C_  (
( _V  X.  _V )  X.  _V ) )
64, 5mpan2 654 . . . 4  |-  ( dom 
A  C_  ( _V  X.  _V )  ->  ( dom  A  X.  ran  A
)  C_  ( ( _V  X.  _V )  X. 
_V ) )
73, 6sylan9ss 3362 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  A  C_  ( ( _V  X.  _V )  X.  _V )
)
8 xpss 4983 . . . . . 6  |-  ( ( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
9 sstr 3357 . . . . . 6  |-  ( ( A  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
)  ->  A  C_  ( _V  X.  _V ) )
108, 9mpan2 654 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  A  C_  ( _V  X.  _V ) )
11 df-rel 4886 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1210, 11sylibr 205 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  Rel  A )
13 dmss 5070 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  dom  ( ( _V  X.  _V )  X.  _V )
)
14 vn0 3636 . . . . . 6  |-  _V  =/=  (/)
15 dmxp 5089 . . . . . 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 3395 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  ( _V  X.  _V ) )
1812, 17jca 520 . . 3  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V ) ) )
197, 18impbii 182 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
202, 19bitri 242 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    =/= wne 2600   _Vcvv 2957    C_ wss 3321   (/)c0 3629    X. cxp 4877   dom cdm 4879   ran crn 4880   Rel wrel 4884
This theorem is referenced by:  dftpos3  6498  tpostpos2  6501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890
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