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Theorem bnj155 28422
Description: Technical lemma for bnj153 28423. 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
bnj155.1  |-  ( ps1  <->  [. g  /  f ]. ps' )
bnj155.2  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj155  |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) )
Distinct variable groups:    A, f    R, f    f, g, i, y
Allowed substitution hints:    A( y, g, i)    R( y, g, i)    ps'( y, f, g, i)    ps1( y, f, g, i)

Proof of Theorem bnj155
StepHypRef Expression
1 bnj155.1 . 2  |-  ( ps1  <->  [. g  /  f ]. ps' )
2 bnj155.2 . . 3  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
32sbcbii 3080 . 2  |-  ( [. g  /  f ]. ps'  <->  [. g  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
4 vex 2825 . . 3  |-  g  e. 
_V
5 fveq1 5562 . . . . . 6  |-  ( f  =  g  ->  (
f `  suc  i )  =  ( g `  suc  i ) )
6 fveq1 5562 . . . . . . 7  |-  ( f  =  g  ->  (
f `  i )  =  ( g `  i ) )
76bnj1113 28328 . . . . . 6  |-  ( f  =  g  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( g `  i
)  pred ( y ,  A ,  R ) )
85, 7eqeq12d 2330 . . . . 5  |-  ( f  =  g  ->  (
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) )
98imbi2d 307 . . . 4  |-  ( f  =  g  ->  (
( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  1o  ->  ( g `  suc  i
)  =  U_ y  e.  ( g `  i
)  pred ( y ,  A ,  R ) ) ) )
109ralbidv 2597 . . 3  |-  ( f  =  g  ->  ( A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) ) )
114, 10sbcie 3059 . 2  |-  ( [. g  /  f ]. A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) )
121, 3, 113bitri 262 1  |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   A.wral 2577   [.wsbc 3025   U_ciun 3942   suc csuc 4431   omcom 4693   ` cfv 5292   1oc1o 6514    predc-bnj14 28224
This theorem is referenced by:  bnj153  28423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-v 2824  df-sbc 3026  df-in 3193  df-ss 3200  df-uni 3865  df-iun 3944  df-br 4061  df-iota 5256  df-fv 5300
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