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Theorem sscoid 30686
Description: A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
sscoid  |-  ( A 
C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B
) )

Proof of Theorem sscoid
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5352 . . 3  |-  Rel  (  _I  o.  B )
2 relss 4941 . . 3  |-  ( A 
C_  (  _I  o.  B )  ->  ( Rel  (  _I  o.  B )  ->  Rel  A ) )
31, 2mpi 20 . 2  |-  ( A 
C_  (  _I  o.  B )  ->  Rel  A )
4 elrel 4956 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  E. y E. z  x  =  <. y ,  z >.
)
5 vex 3083 . . . . . . . . . . 11  |-  y  e. 
_V
6 vex 3083 . . . . . . . . . . 11  |-  z  e. 
_V
75, 6brco 5024 . . . . . . . . . 10  |-  ( y (  _I  o.  B
) z  <->  E. x
( y B x  /\  x  _I  z
) )
8 ancom 451 . . . . . . . . . . . . 13  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  _I  z  /\  y B x ) )
96ideq 5006 . . . . . . . . . . . . . 14  |-  ( x  _I  z  <->  x  =  z )
109anbi1i 699 . . . . . . . . . . . . 13  |-  ( ( x  _I  z  /\  y B x )  <->  ( x  =  z  /\  y B x ) )
118, 10bitri 252 . . . . . . . . . . . 12  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  =  z  /\  y B x ) )
1211exbii 1712 . . . . . . . . . . 11  |-  ( E. x ( y B x  /\  x  _I  z )  <->  E. x
( x  =  z  /\  y B x ) )
13 breq2 4427 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
y B x  <->  y B
z ) )
146, 13ceqsexv 3118 . . . . . . . . . . 11  |-  ( E. x ( x  =  z  /\  y B x )  <->  y B
z )
1512, 14bitri 252 . . . . . . . . . 10  |-  ( E. x ( y B x  /\  x  _I  z )  <->  y B
z )
167, 15bitri 252 . . . . . . . . 9  |-  ( y (  _I  o.  B
) z  <->  y B
z )
1716a1i 11 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( y (  _I  o.  B ) z  <->  y B z ) )
18 eleq1 2495 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  <. y ,  z
>.  e.  (  _I  o.  B ) ) )
19 df-br 4424 . . . . . . . . 9  |-  ( y (  _I  o.  B
) z  <->  <. y ,  z >.  e.  (  _I  o.  B ) )
2018, 19syl6bbr 266 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  y (  _I  o.  B ) z ) )
21 eleq1 2495 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  <. y ,  z
>.  e.  B ) )
22 df-br 4424 . . . . . . . . 9  |-  ( y B z  <->  <. y ,  z >.  e.  B
)
2321, 22syl6bbr 266 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  y B z ) )
2417, 20, 233bitr4d 288 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  x  e.  B
) )
2524exlimivv 1771 . . . . . 6  |-  ( E. y E. z  x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  x  e.  B
) )
264, 25syl 17 . . . . 5  |-  ( ( Rel  A  /\  x  e.  A )  ->  (
x  e.  (  _I  o.  B )  <->  x  e.  B ) )
2726pm5.74da 691 . . . 4  |-  ( Rel 
A  ->  ( (
x  e.  A  ->  x  e.  (  _I  o.  B ) )  <->  ( x  e.  A  ->  x  e.  B ) ) )
2827albidv 1761 . . 3  |-  ( Rel 
A  ->  ( A. x ( x  e.  A  ->  x  e.  (  _I  o.  B
) )  <->  A. x
( x  e.  A  ->  x  e.  B ) ) )
29 dfss2 3453 . . 3  |-  ( A 
C_  (  _I  o.  B )  <->  A. x
( x  e.  A  ->  x  e.  (  _I  o.  B ) ) )
30 dfss2 3453 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
3128, 29, 303bitr4g 291 . 2  |-  ( Rel 
A  ->  ( A  C_  (  _I  o.  B
)  <->  A  C_  B ) )
323, 31biadan2 646 1  |-  ( A 
C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872    C_ wss 3436   <.cop 4004   class class class wbr 4423    _I cid 4763    o. ccom 4857   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-co 4862
This theorem is referenced by:  dffun10  30687
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