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Theorem trrelind 36328
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
StepHypRef Expression
1 trrelind.r . . . 4  |-  ( ph  ->  ( R  o.  R
)  C_  R )
2 inss1 3643 . . . . 5  |-  ( R  i^i  S )  C_  R
32a1i 11 . . . 4  |-  ( ph  ->  ( R  i^i  S
)  C_  R )
41, 3, 3trrelssd 13112 . . 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 3644 . . . . 5  |-  ( R  i^i  S )  C_  S
76a1i 11 . . . 4  |-  ( ph  ->  ( R  i^i  S
)  C_  S )
85, 7, 7trrelssd 13112 . . 3  |-  ( ph  ->  ( ( R  i^i  S )  o.  ( R  i^i  S ) ) 
C_  S )
94, 8ssind 3647 . 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 ) )
1110, 10coeq12d 5004 . 2  |-  ( ph  ->  ( T  o.  T
)  =  ( ( R  i^i  S )  o.  ( R  i^i  S ) ) )
129, 11, 103sstr4d 3461 1  |-  ( ph  ->  ( T  o.  T
)  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    i^i cin 3389    C_ wss 3390    o. ccom 4843
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-in 3397  df-ss 3404  df-br 4396  df-opab 4455  df-co 4848
This theorem is referenced by:  xpintrreld  36329
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