Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1112 Structured version   Unicode version

Theorem bnj1112 32993
Description: Technical lemma for bnj69 33020. 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 32733 . 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 2532 . . . . 5  |-  ( i  =  j  ->  (
i  e.  om  <->  j  e.  om ) )
4 suceq 4936 . . . . . 6  |-  ( i  =  j  ->  suc  i  =  suc  j )
54eleq1d 2529 . . . . 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 5861 . . . . 5  |-  ( i  =  j  ->  (
f `  suc  i )  =  ( f `  suc  j ) )
8 fveq2 5857 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
98bnj1113 32798 . . . . 5  |-  ( i  =  j  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
107, 9eqeq12d 2482 . . . 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 1989 . 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 1372    = wceq 1374    e. wcel 1762   A.wral 2807   U_ciun 4318   suc csuc 4873   ` cfv 5579   omcom 6671    predc-bnj14 32695
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 1961  ax-ext 2438
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 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-suc 4877  df-iota 5542  df-fv 5587
This theorem is referenced by:  bnj1118  32994
  Copyright terms: Public domain W3C validator