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Theorem sspred 5407
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 3687 . 2  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
2 df-pred 5399 . . . 4  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
32sseq1i 3494 . . 3  |-  ( Pred ( R ,  A ,  X )  C_  B  <->  ( A  i^i  ( `' R " { X } ) )  C_  B )
4 df-ss 3456 . . 3  |-  ( ( A  i^i  ( `' R " { X } ) )  C_  B 
<->  ( ( A  i^i  ( `' R " { X } ) )  i^i 
B )  =  ( A  i^i  ( `' R " { X } ) ) )
5 in32 3680 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  i^i 
B )  =  ( ( A  i^i  B
)  i^i  ( `' R " { X }
) )
65eqeq1i 2436 . . 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 274 . 2  |-  ( Pred ( R ,  A ,  X )  C_  B  <->  ( ( A  i^i  B
)  i^i  ( `' R " { X }
) )  =  ( A  i^i  ( `' R " { X } ) ) )
8 ineq1 3663 . . . . 5  |-  ( ( A  i^i  B )  =  B  ->  (
( A  i^i  B
)  i^i  ( `' R " { X }
) )  =  ( B  i^i  ( `' R " { X } ) ) )
98eqeq1d 2431 . . . 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 486 . . 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 5399 . . . . . 6  |-  Pred ( R ,  B ,  X )  =  ( B  i^i  ( `' R " { X } ) )
122, 11eqeq12i 2449 . . . . 5  |-  ( Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X )  <->  ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) ) )
1312biimpri 209 . . . 4  |-  ( ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' R " { X } ) )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
1413eqcoms 2441 . . 3  |-  ( ( B  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' R " { X } ) )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
1510, 14syl 17 . 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 481 1  |-  ( ( B  C_  A  /\  Pred ( R ,  A ,  X )  C_  B
)  ->  Pred ( R ,  A ,  X
)  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    i^i cin 3441    C_ wss 3442   {csn 4002   `'ccnv 4853   "cima 4857   Predcpred 5398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-in 3449  df-ss 3456  df-pred 5399
This theorem is referenced by:  frmin  30267
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