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Theorem predreseq 28836
Description: Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)
Hypothesis
Ref Expression
predreseq.1  |-  X  e. 
_V
Assertion
Ref Expression
predreseq  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( F  |`  Pred ( R ,  A ,  X ) )  =  ( G  |`  Pred ( R ,  A ,  X ) )  <->  A. y  e.  A  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
Distinct variable groups:    y, A    y, F    y, G    y, X    y, R

Proof of Theorem predreseq
StepHypRef Expression
1 predss 28828 . . 3  |-  Pred ( R ,  A ,  X )  C_  A
2 fvreseq 5981 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  Pred ( R ,  A ,  X
)  C_  A )  ->  ( ( F  |`  Pred ( R ,  A ,  X ) )  =  ( G  |`  Pred ( R ,  A ,  X ) )  <->  A. y  e.  Pred  ( R ,  A ,  X )
( F `  y
)  =  ( G `
 y ) ) )
31, 2mpan2 671 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( F  |`  Pred ( R ,  A ,  X ) )  =  ( G  |`  Pred ( R ,  A ,  X ) )  <->  A. y  e.  Pred  ( R ,  A ,  X )
( F `  y
)  =  ( G `
 y ) ) )
4 df-ral 2819 . . . 4  |-  ( A. y  e.  Pred  ( R ,  A ,  X
) ( F `  y )  =  ( G `  y )  <->  A. y ( y  e. 
Pred ( R ,  A ,  X )  ->  ( F `  y
)  =  ( G `
 y ) ) )
5 predreseq.1 . . . . . . 7  |-  X  e. 
_V
6 vex 3116 . . . . . . . 8  |-  y  e. 
_V
76elpred 28834 . . . . . . 7  |-  ( X  e.  _V  ->  (
y  e.  Pred ( R ,  A ,  X )  <->  ( y  e.  A  /\  y R X ) ) )
85, 7ax-mp 5 . . . . . 6  |-  ( y  e.  Pred ( R ,  A ,  X )  <->  ( y  e.  A  /\  y R X ) )
98imbi1i 325 . . . . 5  |-  ( ( y  e.  Pred ( R ,  A ,  X )  ->  ( F `  y )  =  ( G `  y ) )  <->  ( (
y  e.  A  /\  y R X )  -> 
( F `  y
)  =  ( G `
 y ) ) )
109albii 1620 . . . 4  |-  ( A. y ( y  e. 
Pred ( R ,  A ,  X )  ->  ( F `  y
)  =  ( G `
 y ) )  <->  A. y ( ( y  e.  A  /\  y R X )  ->  ( F `  y )  =  ( G `  y ) ) )
11 impexp 446 . . . . 5  |-  ( ( ( y  e.  A  /\  y R X )  ->  ( F `  y )  =  ( G `  y ) )  <->  ( y  e.  A  ->  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
1211albii 1620 . . . 4  |-  ( A. y ( ( y  e.  A  /\  y R X )  ->  ( F `  y )  =  ( G `  y ) )  <->  A. y
( y  e.  A  ->  ( y R X  ->  ( F `  y )  =  ( G `  y ) ) ) )
134, 10, 123bitri 271 . . 3  |-  ( A. y  e.  Pred  ( R ,  A ,  X
) ( F `  y )  =  ( G `  y )  <->  A. y ( y  e.  A  ->  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
14 df-ral 2819 . . 3  |-  ( A. y  e.  A  (
y R X  -> 
( F `  y
)  =  ( G `
 y ) )  <->  A. y ( y  e.  A  ->  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
1513, 14bitr4i 252 . 2  |-  ( A. y  e.  Pred  ( R ,  A ,  X
) ( F `  y )  =  ( G `  y )  <->  A. y  e.  A  ( y R X  ->  ( F `  y )  =  ( G `  y ) ) )
163, 15syl6bb 261 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( F  |`  Pred ( R ,  A ,  X ) )  =  ( G  |`  Pred ( R ,  A ,  X ) )  <->  A. y  e.  A  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    |` cres 5001    Fn wfn 5581   ` cfv 5586   Predcpred 28820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-pred 28821
This theorem is referenced by: (None)
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