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Theorem preddowncl 25473
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 2498 . . . . 5  |-  ( y  =  X  ->  (
y  e.  B  <->  X  e.  B ) )
2 predeq3 25445 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  B , 
y )  =  Pred ( R ,  B ,  X ) )
3 predeq3 25445 . . . . . 6  |-  ( y  =  X  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A ,  X ) )
42, 3eqeq12d 2452 . . . . 5  |-  ( y  =  X  ->  ( Pred ( R ,  B ,  y )  = 
Pred ( R ,  A ,  y )  <->  Pred ( R ,  B ,  X )  =  Pred ( R ,  A ,  X ) ) )
51, 4imbi12d 313 . . . 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 309 . . 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 25447 . . . . . 6  |-  ( B 
C_  A  ->  Pred ( R ,  B , 
y )  C_  Pred ( R ,  A , 
y ) )
87ad2antrr 708 . . . . 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 25445 . . . . . . . . . . . 12  |-  ( x  =  y  ->  Pred ( R ,  A ,  x )  =  Pred ( R ,  A , 
y ) )
109sseq1d 3377 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( Pred ( R ,  A ,  x )  C_  B  <->  Pred ( R ,  A ,  y )  C_  B ) )
1110rspccva 3053 . . . . . . . . . 10  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  Pred ( R ,  A ,  y )  C_  B )
1211sseld 3349 . . . . . . . . 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 2961 . . . . . . . . . . 11  |-  y  e. 
_V
1413elpredim 25453 . . . . . . . . . 10  |-  ( z  e.  Pred ( R ,  A ,  y )  ->  z R y )
1514a1i 11 . . . . . . . . 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 521 . . . . . . . 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 2961 . . . . . . . . . . 11  |-  z  e. 
_V
1817elpred 25454 . . . . . . . . . 10  |-  ( y  e.  B  ->  (
z  e.  Pred ( R ,  B , 
y )  <->  ( z  e.  B  /\  z R y ) ) )
1918imbi2d 309 . . . . . . . . 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 454 . . . . . . . 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 225 . . . . . . 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 3356 . . . . . 6  |-  ( ( A. x  e.  B  Pred ( R ,  A ,  x )  C_  B  /\  y  e.  B
)  ->  Pred ( R ,  A ,  y )  C_  Pred ( R ,  B ,  y ) )
2322adantll 696 . . . . 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 3367 . . . 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 425 . . 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 3013 . 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 49 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   class class class wbr 4214   Predcpred 25440
This theorem is referenced by:  wfrlem4  25543  frrlem4  25587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 25441
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