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Theorem preddowncl 24196
Description: A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
Assertion
Ref Expression
preddowncl  |-  ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  ->  ( X  e.  B  ->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) )
Distinct variable groups:    x, A    x, B    x, R
Allowed substitution hint:    X( x)

Proof of Theorem preddowncl
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . . . . 5  |-  ( y  =  X  ->  (
y  e.  B  <->  X  e.  B ) )
2 predeq3 24171 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B ,  X ) )
3 predeq3 24171 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  X ) )
42, 3eqeq12d 2297 . . . . 5  |-  ( y  =  X  ->  ( Pred ( R ,  B ,  y )  = 
Pred ( R ,  A ,  y )  <->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) )
51, 4imbi12d 311 . . . 4  |-  ( y  =  X  ->  (
( y  e.  B  ->  Pred ( R ,  B ,  y )  =  Pred ( R ,  A ,  y )
)  <->  ( X  e.  B  ->  Pred ( R ,  B ,  X
)  =  Pred ( R ,  A ,  X ) ) ) )
65imbi2d 307 . . 3  |-  ( y  =  X  ->  (
( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B )  ->  (
y  e.  B  ->  Pred ( R ,  B ,  y )  = 
Pred ( R ,  A ,  y )
) )  <->  ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  ->  ( X  e.  B  ->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) ) ) )
7 predpredss 24172 . . . . . 6  |-  ( B 
C_  A  ->  Pred ( R ,  B , 
y )  C_  Pred ( R ,  A , 
y ) )
87ad2antrr 706 . . . . 5  |-  ( ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  /\  y  e.  B )  ->  Pred ( R ,  B , 
y )  C_  Pred ( R ,  A , 
y ) )
9 predeq3 24171 . . . . . . . . . . . 12  |-  ( x  =  y  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
y ) )
109sseq1d 3205 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( Pred ( R ,  A ,  x )  C_  B  <->  Pred ( R ,  A ,  y )  C_  B ) )
1110rspccva 2883 . . . . . . . . . 10  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  Pred ( R ,  A ,  y )  C_  B )
1211sseld 3179 . . . . . . . . 9  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  ( z  e.  Pred ( R ,  A ,  y )  ->  z  e.  B ) )
13 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
1413elpredim 24176 . . . . . . . . . 10  |-  ( z  e.  Pred ( R ,  A ,  y )  ->  z R y )
1514a1i 10 . . . . . . . . 9  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  ( z  e.  Pred ( R ,  A ,  y )  ->  z R y ) )
1612, 15jcad 519 . . . . . . . 8  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  ( z  e.  Pred ( R ,  A ,  y )  ->  ( z  e.  B  /\  z R y ) ) )
17 vex 2791 . . . . . . . . . . 11  |-  z  e. 
_V
1817elpred 24177 . . . . . . . . . 10  |-  ( y  e.  B  ->  (
z  e.  Pred ( R ,  B , 
y )  <->  ( z  e.  B  /\  z R y ) ) )
1918imbi2d 307 . . . . . . . . 9  |-  ( y  e.  B  ->  (
( z  e.  Pred ( R ,  A , 
y )  ->  z  e.  Pred ( R ,  B ,  y )
)  <->  ( z  e. 
Pred ( R ,  A ,  y )  ->  ( z  e.  B  /\  z R y ) ) ) )
2019adantl 452 . . . . . . . 8  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  ( (
z  e.  Pred ( R ,  A , 
y )  ->  z  e.  Pred ( R ,  B ,  y )
)  <->  ( z  e. 
Pred ( R ,  A ,  y )  ->  ( z  e.  B  /\  z R y ) ) ) )
2116, 20mpbird 223 . . . . . . 7  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  ( z  e.  Pred ( R ,  A ,  y )  ->  z  e.  Pred ( R ,  B , 
y ) ) )
2221ssrdv 3185 . . . . . 6  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  Pred ( R ,  A ,  y )  C_  Pred ( R ,  B ,  y ) )
2322adantll 694 . . . . 5  |-  ( ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  /\  y  e.  B )  ->  Pred ( R ,  A , 
y )  C_  Pred ( R ,  B , 
y ) )
248, 23eqssd 3196 . . . 4  |-  ( ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  /\  y  e.  B )  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  A , 
y ) )
2524ex 423 . . 3  |-  ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  ->  ( y  e.  B  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  A , 
y ) ) )
266, 25vtoclg 2843 . 2  |-  ( X  e.  B  ->  (
( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  ->  ( X  e.  B  ->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) ) )
2726pm2.43b 46 1  |-  ( ( B  C_  A  /\  A. x  e.  B  Pred ( R ,  A ,  x )  C_  B
)  ->  ( X  e.  B  ->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   Predcpred 24167
This theorem is referenced by:  wfrlem4  24259  frrlem4  24284
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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