Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj222 Structured version   Unicode version

Theorem bnj222 31881
Description: Technical lemma for bnj229 31882. 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
bnj222.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj222  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, i, m    i, F, m, y   
i, N, m    R, i, m
Allowed substitution hints:    ps( y, i, m)    A( y)    R( y)    N( y)

Proof of Theorem bnj222
StepHypRef Expression
1 bnj222.1 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
2 suceq 4789 . . . . 5  |-  ( i  =  m  ->  suc  i  =  suc  m )
32eleq1d 2509 . . . 4  |-  ( i  =  m  ->  ( suc  i  e.  N  <->  suc  m  e.  N ) )
42fveq2d 5700 . . . . 5  |-  ( i  =  m  ->  ( F `  suc  i )  =  ( F `  suc  m ) )
5 fveq2 5696 . . . . . 6  |-  ( i  =  m  ->  ( F `  i )  =  ( F `  m ) )
65bnj1113 31784 . . . . 5  |-  ( i  =  m  ->  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
74, 6eqeq12d 2457 . . . 4  |-  ( i  =  m  ->  (
( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  <->  ( F `  suc  m )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
83, 7imbi12d 320 . . 3  |-  ( i  =  m  ->  (
( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  m  e.  N  ->  ( F `  suc  m
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) ) )
98cbvralv 2952 . 2  |-  ( A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  <->  A. m  e.  om  ( suc  m  e.  N  ->  ( F `
 suc  m )  =  U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
101, 9bitri 249 1  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2720   U_ciun 4176   suc csuc 4726   ` cfv 5423   omcom 6481    predc-bnj14 31681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-suc 4730  df-iota 5386  df-fv 5431
This theorem is referenced by:  bnj229  31882  bnj589  31907
  Copyright terms: Public domain W3C validator