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Theorem predreseq 24737
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 24731 . . 3  |-  Pred ( R ,  A ,  X )  C_  A
2 fvreseq 5708 . . 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 652 . 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 2624 . . . 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 2867 . . . . . . . 8  |-  y  e. 
_V
76elpred 24735 . . . . . . 7  |-  ( X  e.  _V  ->  (
y  e.  Pred ( R ,  A ,  X )  <->  ( y  e.  A  /\  y R X ) ) )
85, 7ax-mp 8 . . . . . 6  |-  ( y  e.  Pred ( R ,  A ,  X )  <->  ( y  e.  A  /\  y R X ) )
98imbi1i 315 . . . . 5  |-  ( ( y  e.  Pred ( R ,  A ,  X )  ->  ( F `  y )  =  ( G `  y ) )  <->  ( (
y  e.  A  /\  y R X )  -> 
( F `  y
)  =  ( G `
 y ) ) )
109albii 1566 . . . 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 433 . . . . 5  |-  ( ( ( y  e.  A  /\  y R X )  ->  ( F `  y )  =  ( G `  y ) )  <->  ( y  e.  A  ->  ( y R X  ->  ( F `
 y )  =  ( G `  y
) ) ) )
1211albii 1566 . . . 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 262 . . 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 2624 . . 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 243 . 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 252 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228   class class class wbr 4102    |` cres 4770    Fn wfn 5329   ` cfv 5334   Predcpred 24725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342  df-pred 24726
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