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Theorem sscoid 29137
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 5503 . . 3  |-  Rel  (  _I  o.  B )
2 relss 5088 . . 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 5103 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  E. y E. z  x  =  <. y ,  z >.
)
5 vex 3116 . . . . . . . . . . 11  |-  y  e. 
_V
6 vex 3116 . . . . . . . . . . 11  |-  z  e. 
_V
75, 6brco 5171 . . . . . . . . . 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 5153 . . . . . . . . . . . . . 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 1644 . . . . . . . . . . 11  |-  ( E. x ( y B x  /\  x  _I  z )  <->  E. x
( x  =  z  /\  y B x ) )
13 breq2 4451 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
y B x  <->  y B
z ) )
146, 13ceqsexv 3150 . . . . . . . . . . 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 2539 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  <. y ,  z
>.  e.  (  _I  o.  B ) ) )
19 df-br 4448 . . . . . . . . 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 2539 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  <. y ,  z
>.  e.  B ) )
22 df-br 4448 . . . . . . . . 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 1699 . . . . . 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 1689 . . 3  |-  ( Rel 
A  ->  ( A. x ( x  e.  A  ->  x  e.  (  _I  o.  B
) )  <->  A. x
( x  e.  A  ->  x  e.  B ) ) )
29 dfss2 3493 . . 3  |-  ( A 
C_  (  _I  o.  B )  <->  A. x
( x  e.  A  ->  x  e.  (  _I  o.  B ) ) )
30 dfss2 3493 . . 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 1377    = wceq 1379   E.wex 1596    e. wcel 1767    C_ wss 3476   <.cop 4033   class class class wbr 4447    _I cid 4790    o. ccom 5003   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-co 5008
This theorem is referenced by:  dffun10  29138
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