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Theorem bnj1529 33082
Description: Technical lemma for bnj1522 33084. 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 1678 . . 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 2614 . . . . 5  |-  F/_ x F
5 nfcv 2624 . . . . 5  |-  F/_ x
y
64, 5nffv 5866 . . . 4  |-  F/_ x
( F `  y
)
7 nfcv 2624 . . . . 5  |-  F/_ x G
8 nfcv 2624 . . . . . . 7  |-  F/_ x  pred ( y ,  A ,  R )
94, 8nfres 5268 . . . . . 6  |-  F/_ x
( F  |`  pred (
y ,  A ,  R ) )
105, 9nfop 4224 . . . . 5  |-  F/_ x <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
117, 10nffv 5866 . . . 4  |-  F/_ x
( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)
126, 11nfeq 2635 . . 3  |-  F/ x
( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
13 fveq2 5859 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
14 id 22 . . . . . 6  |-  ( x  =  y  ->  x  =  y )
15 bnj602 32929 . . . . . . 7  |-  ( x  =  y  ->  pred (
x ,  A ,  R )  =  pred ( y ,  A ,  R ) )
1615reseq2d 5266 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  pred ( x ,  A ,  R ) )  =  ( F  |`  pred ( y ,  A ,  R ) ) )
1714, 16opeq12d 4216 . . . . 5  |-  ( x  =  y  ->  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.  =  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
1817fveq2d 5863 . . . 4  |-  ( x  =  y  ->  ( G `  <. x ,  ( F  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
1913, 18eqeq12d 2484 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) ) )
202, 12, 19cbvral 3079 . 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 1372    = wceq 1374    e. wcel 1762   A.wral 2809   <.cop 4028    |` cres 4996   ` cfv 5581    predc-bnj14 32697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-res 5006  df-iota 5544  df-fv 5589  df-bnj14 32698
This theorem is referenced by:  bnj1523  33083
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