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Theorem relrelss 4944
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 4435 . . 3  |-  ( Rel 
dom  A  <->  dom  A  C_  ( _V  X.  _V ) )
21anbi2i 445 . 2  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V )
) )
3 relssdmrn 4938 . . . 4  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
4 ssv 3044 . . . . 5  |-  ran  A  C_ 
_V
5 xpss12 4533 . . . . 5  |-  ( ( dom  A  C_  ( _V  X.  _V )  /\  ran  A  C_  _V )  ->  ( dom  A  X.  ran  A )  C_  (
( _V  X.  _V )  X.  _V ) )
64, 5mpan2 416 . . . 4  |-  ( dom 
A  C_  ( _V  X.  _V )  ->  ( dom  A  X.  ran  A
)  C_  ( ( _V  X.  _V )  X. 
_V ) )
73, 6sylan9ss 3036 . . 3  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  ->  A  C_  ( ( _V  X.  _V )  X.  _V )
)
8 xpss 4534 . . . . . 6  |-  ( ( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
9 sstr 3031 . . . . . 6  |-  ( ( A  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( _V  X.  _V )
)  ->  A  C_  ( _V  X.  _V ) )
108, 9mpan2 416 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  A  C_  ( _V  X.  _V ) )
11 df-rel 4435 . . . . 5  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1210, 11sylibr 132 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  Rel  A )
13 dmss 4623 . . . . 5  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  dom  ( ( _V  X.  _V )  X.  _V )
)
14 vn0m 3292 . . . . . 6  |-  E. x  x  e.  _V
15 dmxpm 4644 . . . . . 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 3070 . . . 4  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  dom  A  C_  ( _V  X.  _V ) )
1812, 17jca 300 . . 3  |-  ( A 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  ( Rel  A  /\  dom  A  C_  ( _V  X.  _V ) ) )
197, 18impbii 124 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  ( _V  X.  _V ) )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
202, 19bitri 182 1  |-  ( ( Rel  A  /\  Rel  dom 
A )  <->  A  C_  (
( _V  X.  _V )  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    C_ wss 2997    X. cxp 4426   dom cdm 4428   ran crn 4429   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  dftpos3  6009  tpostpos2  6012
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