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Theorem bnj1112 33772
Description: Technical lemma for bnj69 33799. 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.)
Hypothesis
Ref Expression
bnj1112.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj1112  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, i,
j    R, i, j    f,
i, j, y    i, n, j
Allowed substitution hints:    ps( y, f, i, j, n)    A( y, f, n)    R( y,
f, n)

Proof of Theorem bnj1112
StepHypRef Expression
1 bnj1112.1 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
21bnj115 33511 . 2  |-  ( ps  <->  A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 eleq1 2515 . . . . 5  |-  ( i  =  j  ->  (
i  e.  om  <->  j  e.  om ) )
4 suceq 4933 . . . . . 6  |-  ( i  =  j  ->  suc  i  =  suc  j )
54eleq1d 2512 . . . . 5  |-  ( i  =  j  ->  ( suc  i  e.  n  <->  suc  j  e.  n ) )
63, 5anbi12d 710 . . . 4  |-  ( i  =  j  ->  (
( i  e.  om  /\ 
suc  i  e.  n
)  <->  ( j  e. 
om  /\  suc  j  e.  n ) ) )
74fveq2d 5860 . . . . 5  |-  ( i  =  j  ->  (
f `  suc  i )  =  ( f `  suc  j ) )
8 fveq2 5856 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
98bnj1113 33577 . . . . 5  |-  ( i  =  j  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
107, 9eqeq12d 2465 . . . 4  |-  ( i  =  j  ->  (
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
116, 10imbi12d 320 . . 3  |-  ( i  =  j  ->  (
( ( i  e. 
om  /\  suc  i  e.  n )  ->  (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )  <->  ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
1211cbvalv 2009 . 2  |-  ( A. i ( ( i  e.  om  /\  suc  i  e.  n )  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. j
( ( j  e. 
om  /\  suc  j  e.  n )  ->  (
f `  suc  j )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) )
132, 12bitri 249 1  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   A.wral 2793   U_ciun 4315   suc csuc 4870   ` cfv 5578   omcom 6685    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-iun 4317  df-br 4438  df-suc 4874  df-iota 5541  df-fv 5586
This theorem is referenced by:  bnj1118  33773
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