Theorem sspred 5688

Description: Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Assertion

Ref	Expression
sspred	|- ( ( B C_ A /\ Pred ( R , A , X ) C_ B ) -> Pred ( R , A , X ) = Pred ( R , B , X ) )

Proof of Theorem sspred

Step	Hyp	Ref	Expression
1		sseqin2 3817	. 2 |- ( B C_ A <-> ( A i^i B ) = B )
2		df-pred 5680	. . . 4 |- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
3	2	sseq1i 3629	. . 3 |- ( Pred ( R , A , X ) C_ B <-> ( A i^i ( `' R " { X } ) ) C_ B )
4		df-ss 3588	. . 3 |- ( ( A i^i ( `' R " { X } ) ) C_ B <-> ( ( A i^i ( `' R " { X } ) ) i^i B ) = ( A i^i ( `' R " { X } ) ) )
5		in32 3825	. . . 4 |- ( ( A i^i ( `' R " { X } ) ) i^i B ) = ( ( A i^i B ) i^i ( `' R " { X } ) )
6	5	eqeq1i 2627	. . 3 |- ( ( ( A i^i ( `' R " { X } ) ) i^i B ) = ( A i^i ( `' R " { X } ) ) <-> ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) )
7	3, 4, 6	3bitri 286	. 2 |- ( Pred ( R , A , X ) C_ B <-> ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) )
8		ineq1 3807	. . . . . 6 |- ( ( A i^i B ) = B -> ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( B i^i ( `' R " { X } ) ) )
9	8	eqeq1d 2624	. . . . 5 |- ( ( A i^i B ) = B -> ( ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) <-> ( B i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) ) )
10	9	biimpa 501	. . . 4 |- ( ( ( A i^i B ) = B /\ ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) ) -> ( B i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) )
11		df-pred 5680	. . . 4 |- Pred ( R , B , X ) = ( B i^i ( `' R " { X } ) )
12	10, 11, 2	3eqtr4g 2681	. . 3 |- ( ( ( A i^i B ) = B /\ ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) ) -> Pred ( R , B , X ) = Pred ( R , A , X ) )
13	12	eqcomd 2628	. 2 |- ( ( ( A i^i B ) = B /\ ( ( A i^i B ) i^i ( `' R " { X } ) ) = ( A i^i ( `' R " { X } ) ) ) -> Pred ( R , A , X ) = Pred ( R , B , X ) )
14	1, 7, 13	syl2anb 496	1 |- ( ( B C_ A /\ Pred ( R , A , X ) C_ B ) -> Pred ( R , A , X ) = Pred ( R , B , X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   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:  frmin  31739