Theorem trrelind 37957

Description: The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)

Hypotheses

Ref	Expression
trrelind.r	|- ( ph -> ( R o. R ) C_ R )
trrelind.s	|- ( ph -> ( S o. S ) C_ S )
trrelind.t	|- ( ph -> T = ( R i^i S ) )

Assertion

Ref	Expression
trrelind	|- ( ph -> ( T o. T ) C_ T )

Proof of Theorem trrelind

Step	Hyp	Ref	Expression
1		trrelind.r	. . . 4 |- ( ph -> ( R o. R ) C_ R )
2		inss1 3833	. . . . 5 |- ( R i^i S ) C_ R
3	2	a1i 11	. . . 4 |- ( ph -> ( R i^i S ) C_ R )
4	1, 3, 3	trrelssd 13712	. . 3 |- ( ph -> ( ( R i^i S ) o. ( R i^i S ) ) C_ R )
5		trrelind.s	. . . 4 |- ( ph -> ( S o. S ) C_ S )
6		inss2 3834	. . . . 5 |- ( R i^i S ) C_ S
7	6	a1i 11	. . . 4 |- ( ph -> ( R i^i S ) C_ S )
8	5, 7, 7	trrelssd 13712	. . 3 |- ( ph -> ( ( R i^i S ) o. ( R i^i S ) ) C_ S )
9	4, 8	ssind 3837	. 2 |- ( ph -> ( ( R i^i S ) o. ( R i^i S ) ) C_ ( R i^i S ) )
10		trrelind.t	. . 3 |- ( ph -> T = ( R i^i S ) )
11	10, 10	coeq12d 5286	. 2 |- ( ph -> ( T o. T ) = ( ( R i^i S ) o. ( R i^i S ) ) )
12	9, 11, 10	3sstr4d 3648	1 |- ( ph -> ( T o. T ) C_ T )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   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-v 3202  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  xpintrreld  37958