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Theorem preddowncl 24267
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 2356 . . . . 5  |-  ( y  =  X  ->  (
y  e.  B  <->  X  e.  B ) )
2 predeq3 24242 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B ,  X ) )
3 predeq3 24242 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  X ) )
42, 3eqeq12d 2310 . . . . 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 24243 . . . . . 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 24242 . . . . . . . . . . . 12  |-  ( x  =  y  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
y ) )
109sseq1d 3218 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( Pred ( R ,  A ,  x )  C_  B  <->  Pred ( R ,  A ,  y )  C_  B ) )
1110rspccva 2896 . . . . . . . . . 10  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  Pred ( R ,  A ,  y )  C_  B )
1211sseld 3192 . . . . . . . . 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 2804 . . . . . . . . . . 11  |-  y  e. 
_V
1413elpredim 24247 . . . . . . . . . 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 2804 . . . . . . . . . . 11  |-  z  e. 
_V
1817elpred 24248 . . . . . . . . . 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 3198 . . . . . 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 3209 . . . 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 2856 . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   Predcpred 24238
This theorem is referenced by:  wfrlem4  24330  frrlem4  24355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-pred 24239
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