Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj106 Structured version   Unicode version

Theorem bnj106 29176
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj106.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj106.2  |-  F  e. 
_V
Assertion
Ref Expression
bnj106  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Distinct variable groups:    A, f, n    f, F, i, y    R, f, n    i, n, y
Allowed substitution hints:    ps( y, f, i, n)    A( y,
i)    R( y, i)    F( n)

Proof of Theorem bnj106
StepHypRef Expression
1 bnj106.1 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2 bnj105 29026 . . . 4  |-  1o  e.  _V
31, 2bnj92 29170 . . 3  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
43sbcbii 3208 . 2  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  [. F  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
5 bnj106.2 . . 3  |-  F  e. 
_V
6 fveq1 5719 . . . . . 6  |-  ( f  =  F  ->  (
f `  suc  i )  =  ( F `  suc  i ) )
7 fveq1 5719 . . . . . . 7  |-  ( f  =  F  ->  (
f `  i )  =  ( F `  i ) )
87bnj1113 29093 . . . . . 6  |-  ( f  =  F  ->  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
96, 8eqeq12d 2449 . . . . 5  |-  ( f  =  F  ->  (
( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R )  <->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
109imbi2d 308 . . . 4  |-  ( f  =  F  ->  (
( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  ( suc  i  e.  1o  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) ) ) )
1110ralbidv 2717 . . 3  |-  ( f  =  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  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) ) )
125, 11sbcie 3187 . 2  |-  ( [. F  /  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  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
134, 12bitri 241 1  |-  ( [. F  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153   U_ciun 4085   suc csuc 4575   omcom 4837   ` cfv 5446   1oc1o 6709    predc-bnj14 28989
This theorem is referenced by:  bnj126  29181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812  df-uni 4008  df-iun 4087  df-br 4205  df-suc 4579  df-iota 5410  df-fv 5454  df-1o 6716
  Copyright terms: Public domain W3C validator