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Theorem bnj1529 33859
Description: Technical lemma for bnj1522 33861. 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 1694 . . 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 2595 . . . . 5  |-  F/_ x F
5 nfcv 2605 . . . . 5  |-  F/_ x
y
64, 5nffv 5863 . . . 4  |-  F/_ x
( F `  y
)
7 nfcv 2605 . . . . 5  |-  F/_ x G
8 nfcv 2605 . . . . . . 7  |-  F/_ x  pred ( y ,  A ,  R )
94, 8nfres 5265 . . . . . 6  |-  F/_ x
( F  |`  pred (
y ,  A ,  R ) )
105, 9nfop 4218 . . . . 5  |-  F/_ x <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
117, 10nffv 5863 . . . 4  |-  F/_ x
( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)
126, 11nfeq 2616 . . 3  |-  F/ x
( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
13 fveq2 5856 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
14 id 22 . . . . . 6  |-  ( x  =  y  ->  x  =  y )
15 bnj602 33706 . . . . . . 7  |-  ( x  =  y  ->  pred (
x ,  A ,  R )  =  pred ( y ,  A ,  R ) )
1615reseq2d 5263 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  pred ( x ,  A ,  R ) )  =  ( F  |`  pred ( y ,  A ,  R ) ) )
1714, 16opeq12d 4210 . . . . 5  |-  ( x  =  y  ->  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.  =  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
1817fveq2d 5860 . . . 4  |-  ( x  =  y  ->  ( G `  <. x ,  ( F  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
1913, 18eqeq12d 2465 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) ) )
202, 12, 19cbvral 3066 . 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 1381    = wceq 1383    e. wcel 1804   A.wral 2793   <.cop 4020    |` cres 4991   ` cfv 5578    predc-bnj14 33473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-res 5001  df-iota 5541  df-fv 5586  df-bnj14 33474
This theorem is referenced by:  bnj1523  33860
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