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Theorem bnj953 28650
Description: Technical lemma for bnj69 28719. 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 28416 . . 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 28406 . . 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 241 . 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 28430 . . . . 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 1755 . . . . . 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 420 . . . . 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 462 . . . 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 28487 . . . 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 2408 . . . 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 465 . . 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 2406 . . 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 461 . 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 188 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   A.wral 2651   U_ciun 4037   suc csuc 4526   omcom 4787   ` cfv 5396    /\ w-bnj17 28390    predc-bnj14 28392
This theorem is referenced by:  bnj967  28656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-ral 2656  df-rex 2657  df-iun 4039  df-bnj17 28391
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