Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1529 Structured version   Unicode version

Theorem bnj1529 32059
Description: Technical lemma for bnj1522 32061. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1529.1  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
bnj1529.2  |-  ( w  e.  F  ->  A. x  w  e.  F )
Assertion
Ref Expression
bnj1529  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
Distinct variable groups:    w, A, x, y    w, F, y   
w, G, x, y   
w, R, x, y
Allowed substitution hints:    ch( x, y, w)    F( x)

Proof of Theorem bnj1529
StepHypRef Expression
1 bnj1529.1 . 2  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
2 nfv 1673 . . 3  |-  F/ y ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
3 bnj1529.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
43nfcii 2569 . . . . 5  |-  F/_ x F
5 nfcv 2578 . . . . 5  |-  F/_ x
y
64, 5nffv 5697 . . . 4  |-  F/_ x
( F `  y
)
7 nfcv 2578 . . . . 5  |-  F/_ x G
8 nfcv 2578 . . . . . . 7  |-  F/_ x  pred ( y ,  A ,  R )
94, 8nfres 5111 . . . . . 6  |-  F/_ x
( F  |`  pred (
y ,  A ,  R ) )
105, 9nfop 4074 . . . . 5  |-  F/_ x <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
117, 10nffv 5697 . . . 4  |-  F/_ x
( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)
126, 11nfeq 2585 . . 3  |-  F/ x
( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
13 fveq2 5690 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
14 id 22 . . . . . 6  |-  ( x  =  y  ->  x  =  y )
15 bnj602 31906 . . . . . . 7  |-  ( x  =  y  ->  pred (
x ,  A ,  R )  =  pred ( y ,  A ,  R ) )
1615reseq2d 5109 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  pred ( x ,  A ,  R ) )  =  ( F  |`  pred ( y ,  A ,  R ) ) )
1714, 16opeq12d 4066 . . . . 5  |-  ( x  =  y  ->  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.  =  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
1817fveq2d 5694 . . . 4  |-  ( x  =  y  ->  ( G `  <. x ,  ( F  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
1913, 18eqeq12d 2456 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) ) )
202, 12, 19cbvral 2942 . 2  |-  ( A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  <->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
) )
211, 20sylib 196 1  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2714   <.cop 3882    |` cres 4841   ` cfv 5417    predc-bnj14 31674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-xp 4845  df-res 4851  df-iota 5380  df-fv 5425  df-bnj14 31675
This theorem is referenced by:  bnj1523  32060
  Copyright terms: Public domain W3C validator