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Theorem sscoid 28075
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 5431 . . 3  |-  Rel  (  _I  o.  B )
2 relss 5022 . . 3  |-  ( A 
C_  (  _I  o.  B )  ->  ( Rel  (  _I  o.  B )  ->  Rel  A ) )
31, 2mpi 17 . 2  |-  ( A 
C_  (  _I  o.  B )  ->  Rel  A )
4 elrel 5037 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  E. y E. z  x  =  <. y ,  z >.
)
5 vex 3068 . . . . . . . . . . 11  |-  y  e. 
_V
6 vex 3068 . . . . . . . . . . 11  |-  z  e. 
_V
75, 6brco 5105 . . . . . . . . . 10  |-  ( y (  _I  o.  B
) z  <->  E. x
( y B x  /\  x  _I  z
) )
8 ancom 450 . . . . . . . . . . . . 13  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  _I  z  /\  y B x ) )
96ideq 5087 . . . . . . . . . . . . . 14  |-  ( x  _I  z  <->  x  =  z )
109anbi1i 695 . . . . . . . . . . . . 13  |-  ( ( x  _I  z  /\  y B x )  <->  ( x  =  z  /\  y B x ) )
118, 10bitri 249 . . . . . . . . . . . 12  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  =  z  /\  y B x ) )
1211exbii 1635 . . . . . . . . . . 11  |-  ( E. x ( y B x  /\  x  _I  z )  <->  E. x
( x  =  z  /\  y B x ) )
13 breq2 4391 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
y B x  <->  y B
z ) )
146, 13ceqsexv 3102 . . . . . . . . . . 11  |-  ( E. x ( x  =  z  /\  y B x )  <->  y B
z )
1512, 14bitri 249 . . . . . . . . . 10  |-  ( E. x ( y B x  /\  x  _I  z )  <->  y B
z )
167, 15bitri 249 . . . . . . . . 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 2521 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  <. y ,  z
>.  e.  (  _I  o.  B ) ) )
19 df-br 4388 . . . . . . . . 9  |-  ( y (  _I  o.  B
) z  <->  <. y ,  z >.  e.  (  _I  o.  B ) )
2018, 19syl6bbr 263 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  y (  _I  o.  B ) z ) )
21 eleq1 2521 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  <. y ,  z
>.  e.  B ) )
22 df-br 4388 . . . . . . . . 9  |-  ( y B z  <->  <. y ,  z >.  e.  B
)
2321, 22syl6bbr 263 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  y B z ) )
2417, 20, 233bitr4d 285 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  x  e.  B
) )
2524exlimivv 1690 . . . . . 6  |-  ( E. y E. z  x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  x  e.  B
) )
264, 25syl 16 . . . . 5  |-  ( ( Rel  A  /\  x  e.  A )  ->  (
x  e.  (  _I  o.  B )  <->  x  e.  B ) )
2726pm5.74da 687 . . . 4  |-  ( Rel 
A  ->  ( (
x  e.  A  ->  x  e.  (  _I  o.  B ) )  <->  ( x  e.  A  ->  x  e.  B ) ) )
2827albidv 1680 . . 3  |-  ( Rel 
A  ->  ( A. x ( x  e.  A  ->  x  e.  (  _I  o.  B
) )  <->  A. x
( x  e.  A  ->  x  e.  B ) ) )
29 dfss2 3440 . . 3  |-  ( A 
C_  (  _I  o.  B )  <->  A. x
( x  e.  A  ->  x  e.  (  _I  o.  B ) ) )
30 dfss2 3440 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
3128, 29, 303bitr4g 288 . 2  |-  ( Rel 
A  ->  ( A  C_  (  _I  o.  B
)  <->  A  C_  B ) )
323, 31biadan2 642 1  |-  ( A 
C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758    C_ wss 3423   <.cop 3978   class class class wbr 4387    _I cid 4726    o. ccom 4939   Rel wrel 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-co 4944
This theorem is referenced by:  dffun10  28076
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