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Theorem bnj953 29311
Description: Technical lemma for bnj69 29380. 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
bnj953.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj953.2  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
Assertion
Ref Expression
bnj953  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )

Proof of Theorem bnj953
StepHypRef Expression
1 bnj312 29078 . . 3  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  <->  ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( G `  i )  =  ( f `  i )  /\  ( i  e. 
om  /\  suc  i  e.  n )  /\  ps ) )
2 bnj251 29068 . . 3  |-  ( ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( G `  i )  =  ( f `  i )  /\  ( i  e. 
om  /\  suc  i  e.  n )  /\  ps ) 
<->  ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( ( G `  i )  =  ( f `  i )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )
) ) )
31, 2bitri 249 . 2  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  <->  ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( ( G `  i )  =  ( f `  i )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )
) ) )
4 bnj953.1 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
54bnj115 29092 . . . . 5  |-  ( ps  <->  A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
6 sp 1883 . . . . . 6  |-  ( A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  -> 
( ( i  e. 
om  /\  suc  i  e.  n )  ->  (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) )
76impcom 428 . . . . 5  |-  ( ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  A. i
( ( i  e. 
om  /\  suc  i  e.  n )  ->  (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) ) )  ->  (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
85, 7sylan2b 473 . . . 4  |-  ( ( ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )
9 bnj953.2 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
109bnj956 29149 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
11 eqtr3 2430 . . . 4  |-  ( ( ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  /\  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  -> 
( f `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
128, 10, 11syl2anr 476 . . 3  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( ( i  e. 
om  /\  suc  i  e.  n )  /\  ps ) )  ->  (
f `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
13 eqtr 2428 . . 3  |-  ( ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( f `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) )  ->  ( G `  suc  i )  = 
U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) )
1412, 13sylan2 472 . 2  |-  ( ( ( G `  suc  i )  =  ( f `  suc  i
)  /\  ( ( G `  i )  =  ( f `  i )  /\  (
( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
153, 14sylbi 195 1  |-  ( ( ( G `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842   A.wral 2753   U_ciun 4270   suc csuc 5411   ` cfv 5568   omcom 6682    /\ w-bnj17 29052    predc-bnj14 29054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-ral 2758  df-rex 2759  df-iun 4272  df-bnj17 29053
This theorem is referenced by:  bnj967  29317
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