Theorem trclsslem 13729

Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)

Assertion

Ref	Expression
trclsslem	|- ( R C_ S -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ |^| { r | ( S C_ r /\ ( r o. r ) C_ r ) } )

Distinct variable groups:   R, r   S, r

Proof of Theorem trclsslem

Step	Hyp	Ref	Expression
1		clsslem 13723	1 |- ( R C_ S -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ |^| { r | ( S C_ r /\ ( r o. r ) C_ r ) } )

Syntax hints:   -> wi 4   /\ wa 384   {cab 2608   C_ wss 3574   |^|cint 4475   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-ral 2917  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  trclfvss  13747