Theorem bnj1414

Description: Property of _trCl . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1414.1	|- B = ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) )
Assertion	bnj1414	|- ( ( R _FrSe A /\ X e. A ) -> _trCl ( X , A , R ) = B )

Proof

Step	Hyp	Ref	Expression
1		bnj1414.1	|- B = ( _pred ( X , A , R ) u. U_ y e. _pred ( X , A , R ) _trCl ( y , A , R ) )
2		eqid	|- ( _pred ( X , A , R ) u. U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) ) = ( _pred ( X , A , R ) u. U_ y e. _trCl ( X , A , R ) _trCl ( y , A , R ) )
3		biid	|- ( ( R _FrSe A /\ X e. A ) <-> ( R _FrSe A /\ X e. A ) )
4		biid	|- ( ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
5	1 2 3 4	bnj1408	|- ( ( R _FrSe A /\ X e. A ) -> _trCl ( X , A , R ) = B )