Theorem trrelssd 13712

Description: The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)

Hypotheses

Ref	Expression
trrelssd.r	|- ( ph -> ( R o. R ) C_ R )
trrelssd.s	|- ( ph -> S C_ R )
trrelssd.t	|- ( ph -> T C_ R )

Assertion

Ref	Expression
trrelssd	|- ( ph -> ( S o. T ) C_ R )

Proof of Theorem trrelssd

Step	Hyp	Ref	Expression
1		trrelssd.s	. . 3 |- ( ph -> S C_ R )
2		trrelssd.t	. . 3 |- ( ph -> T C_ R )
3	1, 2	coss12d 13711	. 2 |- ( ph -> ( S o. T ) C_ ( R o. R ) )
4		trrelssd.r	. 2 |- ( ph -> ( R o. R ) C_ R )
5	3, 4	sstrd 3613	1 |- ( ph -> ( S o. T ) C_ R )

Syntax hints:   -> wi 4   C_ wss 3574   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:  trclfvlb2  13751  trrelind  37957  iunrelexpmin1  38000  iunrelexpmin2  38004