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Theorem bnj1123 31977
Description: Technical lemma for bnj69 32001. 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
bnj1123.4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1123.3  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1123.1  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
bnj1123.2  |-  ( et'  <->  [. j  /  i ]. et )
Assertion
Ref Expression
bnj1123  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
Distinct variable groups:    B, i    D, i    f, i    i,
j    i, n    ph, i
Allowed substitution hints:    ph( y, f, j, n)    ps( y,
f, i, j, n)    et( y, f, i, j, n)    A( y, f, i, j, n)    B( y,
f, j, n)    D( y, f, j, n)    R( y, f, i, j, n)    K( y, f, i, j, n)    et'( y, f, i, j, n)

Proof of Theorem bnj1123
StepHypRef Expression
1 bnj1123.2 . 2  |-  ( et'  <->  [. j  /  i ]. et )
2 bnj1123.1 . . 3  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
32sbcbii 3246 . 2  |-  ( [. j  /  i ]. et  <->  [. j  /  i ]. ( ( f  e.  K  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)
4 vex 2975 . . 3  |-  j  e. 
_V
5 bnj1123.3 . . . . . . . 8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
6 nfcv 2579 . . . . . . . . . 10  |-  F/_ i D
7 nfv 1673 . . . . . . . . . . 11  |-  F/ i  f  Fn  n
8 nfv 1673 . . . . . . . . . . 11  |-  F/ i
ph
9 bnj1123.4 . . . . . . . . . . . . 13  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
109bnj1095 31775 . . . . . . . . . . . 12  |-  ( ps 
->  A. i ps )
1110nfi 1596 . . . . . . . . . . 11  |-  F/ i ps
127, 8, 11nf3an 1863 . . . . . . . . . 10  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
136, 12nfrex 2771 . . . . . . . . 9  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
1413nfab 2583 . . . . . . . 8  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
155, 14nfcxfr 2576 . . . . . . 7  |-  F/_ i K
1615nfcri 2573 . . . . . 6  |-  F/ i  f  e.  K
17 nfv 1673 . . . . . 6  |-  F/ i  j  e.  dom  f
1816, 17nfan 1861 . . . . 5  |-  F/ i ( f  e.  K  /\  j  e.  dom  f )
19 nfv 1673 . . . . 5  |-  F/ i ( f `  j
)  C_  B
2018, 19nfim 1853 . . . 4  |-  F/ i ( ( f  e.  K  /\  j  e. 
dom  f )  -> 
( f `  j
)  C_  B )
21 eleq1 2503 . . . . . 6  |-  ( i  =  j  ->  (
i  e.  dom  f  <->  j  e.  dom  f ) )
2221anbi2d 703 . . . . 5  |-  ( i  =  j  ->  (
( f  e.  K  /\  i  e.  dom  f )  <->  ( f  e.  K  /\  j  e.  dom  f ) ) )
23 fveq2 5691 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
2423sseq1d 3383 . . . . 5  |-  ( i  =  j  ->  (
( f `  i
)  C_  B  <->  ( f `  j )  C_  B
) )
2522, 24imbi12d 320 . . . 4  |-  ( i  =  j  ->  (
( ( f  e.  K  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) ) )
2620, 25sbciegf 3218 . . 3  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ( ( f  e.  K  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) ) )
274, 26ax-mp 5 . 2  |-  ( [. j  /  i ]. (
( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B )  <->  ( (
f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j )  C_  B
) )
281, 3, 273bitri 271 1  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2715   E.wrex 2716   _Vcvv 2972   [.wsbc 3186    C_ wss 3328   U_ciun 4171   suc csuc 4721   dom cdm 4840    Fn wfn 5413   ` cfv 5418   omcom 6476    predc-bnj14 31676
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 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426
This theorem is referenced by:  bnj1030  31978
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