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Theorem bnj1529 29439
Description: Technical lemma for bnj1522 29441. 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 1629 . . 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 2563 . . . . 5  |-  F/_ x F
5 nfcv 2572 . . . . 5  |-  F/_ x
y
64, 5nffv 5735 . . . 4  |-  F/_ x
( F `  y
)
7 nfcv 2572 . . . . 5  |-  F/_ x G
8 nfcv 2572 . . . . . . 7  |-  F/_ x  pred ( y ,  A ,  R )
94, 8nfres 5148 . . . . . 6  |-  F/_ x
( F  |`  pred (
y ,  A ,  R ) )
105, 9nfop 4000 . . . . 5  |-  F/_ x <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
117, 10nffv 5735 . . . 4  |-  F/_ x
( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)
126, 11nfeq 2579 . . 3  |-  F/ x
( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
13 fveq2 5728 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
14 id 20 . . . . . 6  |-  ( x  =  y  ->  x  =  y )
15 bnj602 29286 . . . . . . 7  |-  ( x  =  y  ->  pred (
x ,  A ,  R )  =  pred ( y ,  A ,  R ) )
1615reseq2d 5146 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  pred ( x ,  A ,  R ) )  =  ( F  |`  pred ( y ,  A ,  R ) ) )
1714, 16opeq12d 3992 . . . . 5  |-  ( x  =  y  ->  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.  =  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
1817fveq2d 5732 . . . 4  |-  ( x  =  y  ->  ( G `  <. x ,  ( F  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
1913, 18eqeq12d 2450 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) ) )
202, 12, 19cbvral 2928 . 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 189 1  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817    |` cres 4880   ` cfv 5454    predc-bnj14 29052
This theorem is referenced by:  bnj1523  29440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-res 4890  df-iota 5418  df-fv 5462  df-bnj14 29053
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