Theorem coss12d 13711

Description: Subset deduction for composition of two classes. (Contributed by RP, 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.

Step	Hyp	Ref	Expression
1		coss12d.c	. . . . . 6 |- ( ph -> C C_ D )
2	1	ssbrd 4696	. . . . 5 |- ( ph -> ( x C y -> x D y ) )
3		coss12d.a	. . . . . 6 |- ( ph -> A C_ B )
4	3	ssbrd 4696	. . . . 5 |- ( ph -> ( y A z -> y B z ) )
5	2, 4	anim12d 586	. . . 4 |- ( ph -> ( ( x C y /\ y A z ) -> ( x D y /\ y B z ) ) )
6	5	eximdv 1846	. . 3 |- ( ph -> ( E. y ( x C y /\ y A z ) -> E. y ( x D y /\ y B z ) ) )
7	6	ssopab2dv 5004	. 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 5123	. 2 |- ( A o. C ) = { <. x , z >. | E. y ( x C y /\ y A z ) }
9		df-co 5123	. 2 |- ( B o. D ) = { <. x , z >. | E. y ( x D y /\ y B z ) }
10	7, 8, 9	3sstr4g 3646	1 |- ( ph -> ( A o. C ) C_ ( B o. D ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   C_ wss 3574   class class class wbr 4653   {copab 4712   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  trrelssd  13712  relexpss1d  37997