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Theorem sscoid 29725
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 5511 . . 3  |-  Rel  (  _I  o.  B )
2 relss 5099 . . 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 5114 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  E. y E. z  x  =  <. y ,  z >.
)
5 vex 3112 . . . . . . . . . . 11  |-  y  e. 
_V
6 vex 3112 . . . . . . . . . . 11  |-  z  e. 
_V
75, 6brco 5183 . . . . . . . . . 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 5165 . . . . . . . . . . . . . 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 1668 . . . . . . . . . . 11  |-  ( E. x ( y B x  /\  x  _I  z )  <->  E. x
( x  =  z  /\  y B x ) )
13 breq2 4460 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
y B x  <->  y B
z ) )
146, 13ceqsexv 3146 . . . . . . . . . . 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 2529 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  <. y ,  z
>.  e.  (  _I  o.  B ) ) )
19 df-br 4457 . . . . . . . . 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 2529 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  <. y ,  z
>.  e.  B ) )
22 df-br 4457 . . . . . . . . 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 1724 . . . . . 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 1714 . . 3  |-  ( Rel 
A  ->  ( A. x ( x  e.  A  ->  x  e.  (  _I  o.  B
) )  <->  A. x
( x  e.  A  ->  x  e.  B ) ) )
29 dfss2 3488 . . 3  |-  ( A 
C_  (  _I  o.  B )  <->  A. x
( x  e.  A  ->  x  e.  (  _I  o.  B ) ) )
30 dfss2 3488 . . 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 1393    = wceq 1395   E.wex 1613    e. wcel 1819    C_ wss 3471   <.cop 4038   class class class wbr 4456    _I cid 4799    o. ccom 5012   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-co 5017
This theorem is referenced by:  dffun10  29726
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