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Theorem bnj540 28435
Description: Technical lemma for bnj852 28464. 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
bnj540.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj540.2  |-  ( ps"  <->  [. G  / 
f ]. ps )
bnj540.3  |-  G  e. 
_V
Assertion
Ref Expression
bnj540  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
Distinct variable groups:    A, f    f, G, i, y    f, N    R, f
Allowed substitution hints:    ps( y, f, i)    A( y, i)    R( y, i)    N( y, i)    ps"( y, f, i)

Proof of Theorem bnj540
StepHypRef Expression
1 bnj540.2 . 2  |-  ( ps"  <->  [. G  / 
f ]. ps )
2 bnj540.1 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj540.3 . . . 4  |-  G  e. 
_V
42, 3bnj524 28277 . . 3  |-  ( [. G  /  f ]. ps  <->  [. G  /  f ]. A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
53bnj538 28280 . . 3  |-  ( [. G  /  f ]. A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  [. G  / 
f ]. ( suc  i  e.  N  ->  ( f `
 suc  i )  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
6 sbcimg 3066 . . . . 5  |-  ( G  e.  _V  ->  ( [. G  /  f ]. ( suc  i  e.  N  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )  <->  ( [. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
73, 6ax-mp 8 . . . 4  |-  ( [. G  /  f ]. ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( [. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
87ralbii 2601 . . 3  |-  ( A. i  e.  om  [. G  /  f ]. ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( [. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
94, 5, 83bitri 262 . 2  |-  ( [. G  /  f ]. ps  <->  A. i  e.  om  ( [. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
103bnj525 28278 . . . 4  |-  ( [. G  /  f ]. suc  i  e.  N  <->  suc  i  e.  N )
11 fveq1 5562 . . . . . 6  |-  ( f  =  G  ->  (
f `  suc  i )  =  ( G `  suc  i ) )
12 fveq1 5562 . . . . . . 7  |-  ( f  =  G  ->  (
f `  i )  =  ( G `  i ) )
1312bnj1113 28328 . . . . . 6  |-  ( f  =  G  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
1411, 13eqeq12d 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 ) ) )
153, 14sbcie 3059 . . . 4  |-  ( [. G  /  f ]. (
f `  suc  i )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  <-> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
1610, 15imbi12i 316 . . 3  |-  ( (
[. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  N  ->  ( G `  suc  i
)  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
1716ralbii 2601 . 2  |-  ( A. i  e.  om  ( [. G  /  f ]. suc  i  e.  N  ->  [. G  /  f ]. ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
181, 9, 173bitri 262 1  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( 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   _Vcvv 2822   [.wsbc 3025   U_ciun 3942   suc csuc 4431   omcom 4693   ` cfv 5292    predc-bnj14 28224
This theorem is referenced by:  bnj580  28456  bnj607  28459
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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