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Theorem coss12d 37462
Description: Subset deduction for composition of two classes. (Contributed by Richard Penner, 24-Dec-2019.)
Hypotheses
Ref Expression
coss12d.a  |-  ( ph  ->  A  C_  B )
coss12d.c  |-  ( ph  ->  C  C_  D )
Assertion
Ref Expression
coss12d  |-  ( ph  ->  ( A  o.  C
)  C_  ( B  o.  D ) )

Proof of Theorem coss12d
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coss12d.c . . . . . 6  |-  ( ph  ->  C  C_  D )
21ssbrd 4478 . . . . 5  |-  ( ph  ->  ( x C y  ->  x D y ) )
3 coss12d.a . . . . . 6  |-  ( ph  ->  A  C_  B )
43ssbrd 4478 . . . . 5  |-  ( ph  ->  ( y A z  ->  y B z ) )
52, 4anim12d 563 . . . 4  |-  ( ph  ->  ( ( x C y  /\  y A z )  ->  (
x D y  /\  y B z ) ) )
65eximdv 1697 . . 3  |-  ( ph  ->  ( E. y ( x C y  /\  y A z )  ->  E. y ( x D y  /\  y B z ) ) )
76ssopab2dv 4766 . 2  |-  ( ph  ->  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }  C_  { <. x ,  z >.  |  E. y ( x D y  /\  y B z ) } )
8 df-co 4998 . 2  |-  ( A  o.  C )  =  { <. x ,  z
>.  |  E. y
( x C y  /\  y A z ) }
9 df-co 4998 . 2  |-  ( B  o.  D )  =  { <. x ,  z
>.  |  E. y
( x D y  /\  y B z ) }
107, 8, 93sstr4g 3530 1  |-  ( ph  ->  ( A  o.  C
)  C_  ( B  o.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1599    C_ wss 3461   class class class wbr 4437   {copab 4494    o. ccom 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-in 3468  df-ss 3475  df-br 4438  df-opab 4496  df-co 4998
This theorem is referenced by:  trrelssd  37470
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