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Theorem sspred 24174
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
StepHypRef Expression
1 sseqin2 3388 . 2  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
2 df-pred 24168 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
32sseq1i 3202 . . 3  |-  ( Pred ( R ,  A ,  X )  C_  B  <->  ( A  i^i  ( `' R " { X } ) )  C_  B )
4 df-ss 3166 . . 3  |-  ( ( A  i^i  ( `' R " { X } ) )  C_  B 
<->  ( ( A  i^i  ( `' R " { X } ) )  i^i 
B )  =  ( A  i^i  ( `' R " { X } ) ) )
5 in32 3381 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i 
B )  =  ( ( A  i^i  B
)  i^i  ( `' R " { X }
) )
65eqeq1i 2290 . . 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 } ) ) )
73, 4, 63bitri 262 . 2  |-  ( Pred ( R ,  A ,  X )  C_  B  <->  ( ( A  i^i  B
)  i^i  ( `' R " { X }
) )  =  ( A  i^i  ( `' R " { X } ) ) )
8 ineq1 3363 . . . . 5  |-  ( ( A  i^i  B )  =  B  ->  (
( A  i^i  B
)  i^i  ( `' R " { X }
) )  =  ( B  i^i  ( `' R " { X } ) ) )
98eqeq1d 2291 . . . 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 } ) ) ) )
109biimpa 470 . . 3  |-  ( ( ( 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 24168 . . . . . 6  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
122, 11eqeq12i 2296 . . . . 5  |-  ( Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X )  <->  ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) ) )
1312biimpri 197 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
1413eqcoms 2286 . . 3  |-  ( ( B  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' R " { X } ) )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
1510, 14syl 15 . 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 )
)
161, 7, 15syl2anb 465 1  |-  ( ( B  C_  A  /\  Pred ( R ,  A ,  X )  C_  B
)  ->  Pred ( R ,  A ,  X
)  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    i^i cin 3151    C_ wss 3152   {csn 3640   `'ccnv 4688   "cima 4692   Predcpred 24167
This theorem is referenced by:  frmin  24242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pred 24168
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