Theorem predpredss 5686

Description: If A is a subset of B, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predpredss	|- ( A C_ B -> Pred ( R , A , X ) C_ Pred ( R , B , X ) )

Proof of Theorem predpredss

Step	Hyp	Ref	Expression
1		ssrin 3838	. 2 |- ( A C_ B -> ( A i^i ( `' R " { X } ) ) C_ ( B i^i ( `' R " { X } ) ) )
2		df-pred 5680	. 2 |- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
3		df-pred 5680	. 2 |- Pred ( R , B , X ) = ( B i^i ( `' R " { X } ) )
4	1, 2, 3	3sstr4g 3646	1 |- ( A C_ B -> Pred ( R , A , X ) C_ Pred ( R , B , X ) )

Syntax hints:   -> wi 4   i^i cin 3573   C_ wss 3574   {csn 4177   `'ccnv 5113   "cima 5117   Predcpred 5679

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-pred 5680

This theorem is referenced by:  preddowncl  5707  wfrlem8  7422