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Theorem sscoid 30751
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 5340 . . 3  |-  Rel  (  _I  o.  B )
2 relss 4927 . . 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 4942 . . . . . 6  |-  ( ( Rel  A  /\  x  e.  A )  ->  E. y E. z  x  =  <. y ,  z >.
)
5 vex 3034 . . . . . . . . . . 11  |-  y  e. 
_V
6 vex 3034 . . . . . . . . . . 11  |-  z  e. 
_V
75, 6brco 5010 . . . . . . . . . 10  |-  ( y (  _I  o.  B
) z  <->  E. x
( y B x  /\  x  _I  z
) )
8 ancom 457 . . . . . . . . . . . . 13  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  _I  z  /\  y B x ) )
96ideq 4992 . . . . . . . . . . . . . 14  |-  ( x  _I  z  <->  x  =  z )
109anbi1i 709 . . . . . . . . . . . . 13  |-  ( ( x  _I  z  /\  y B x )  <->  ( x  =  z  /\  y B x ) )
118, 10bitri 257 . . . . . . . . . . . 12  |-  ( ( y B x  /\  x  _I  z )  <->  ( x  =  z  /\  y B x ) )
1211exbii 1726 . . . . . . . . . . 11  |-  ( E. x ( y B x  /\  x  _I  z )  <->  E. x
( x  =  z  /\  y B x ) )
13 breq2 4399 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
y B x  <->  y B
z ) )
146, 13ceqsexv 3070 . . . . . . . . . . 11  |-  ( E. x ( x  =  z  /\  y B x )  <->  y B
z )
1512, 14bitri 257 . . . . . . . . . 10  |-  ( E. x ( y B x  /\  x  _I  z )  <->  y B
z )
167, 15bitri 257 . . . . . . . . 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 2537 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  <. y ,  z
>.  e.  (  _I  o.  B ) ) )
19 df-br 4396 . . . . . . . . 9  |-  ( y (  _I  o.  B
) z  <->  <. y ,  z >.  e.  (  _I  o.  B ) )
2018, 19syl6bbr 271 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  y (  _I  o.  B ) z ) )
21 eleq1 2537 . . . . . . . . 9  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  <. y ,  z
>.  e.  B ) )
22 df-br 4396 . . . . . . . . 9  |-  ( y B z  <->  <. y ,  z >.  e.  B
)
2321, 22syl6bbr 271 . . . . . . . 8  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  B  <->  y B z ) )
2417, 20, 233bitr4d 293 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  ( x  e.  (  _I  o.  B
)  <->  x  e.  B
) )
2524exlimivv 1786 . . . . . 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 701 . . . 4  |-  ( Rel 
A  ->  ( (
x  e.  A  ->  x  e.  (  _I  o.  B ) )  <->  ( x  e.  A  ->  x  e.  B ) ) )
2827albidv 1775 . . 3  |-  ( Rel 
A  ->  ( A. x ( x  e.  A  ->  x  e.  (  _I  o.  B
) )  <->  A. x
( x  e.  A  ->  x  e.  B ) ) )
29 dfss2 3407 . . 3  |-  ( A 
C_  (  _I  o.  B )  <->  A. x
( x  e.  A  ->  x  e.  (  _I  o.  B ) ) )
30 dfss2 3407 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
3128, 29, 303bitr4g 296 . 2  |-  ( Rel 
A  ->  ( A  C_  (  _I  o.  B
)  <->  A  C_  B ) )
323, 31biadan2 654 1  |-  ( A 
C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904    C_ wss 3390   <.cop 3965   class class class wbr 4395    _I cid 4749    o. ccom 4843   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-co 4848
This theorem is referenced by:  dffun10  30752
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