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Theorem bnj1123 28389
Description: Technical lemma for bnj69 28413. 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 3046 . 2  |-  ( [. j  /  i ]. et  <->  [. j  /  i ]. ( ( f  e.  K  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  B )
)
4 vex 2791 . . 3  |-  j  e. 
_V
5 bnj1123.3 . . . . . . . 8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
6 nfcv 2419 . . . . . . . . . 10  |-  F/_ i D
7 nfv 1605 . . . . . . . . . . 11  |-  F/ i  f  Fn  n
8 nfv 1605 . . . . . . . . . . 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 28186 . . . . . . . . . . . 12  |-  ( ps 
->  A. i ps )
1110nfi 1538 . . . . . . . . . . 11  |-  F/ i ps
127, 8, 11nf3an 1774 . . . . . . . . . 10  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
136, 12nfrex 2598 . . . . . . . . 9  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
1413nfab 2423 . . . . . . . 8  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
155, 14nfcxfr 2416 . . . . . . 7  |-  F/_ i K
1615nfcri 2413 . . . . . 6  |-  F/ i  f  e.  K
17 nfv 1605 . . . . . 6  |-  F/ i  j  e.  dom  f
1816, 17nfan 1771 . . . . 5  |-  F/ i ( f  e.  K  /\  j  e.  dom  f )
19 nfv 1605 . . . . 5  |-  F/ i ( f `  j
)  C_  B
2018, 19nfim 1769 . . . 4  |-  F/ i ( ( f  e.  K  /\  j  e. 
dom  f )  -> 
( f `  j
)  C_  B )
21 eleq1 2343 . . . . . 6  |-  ( i  =  j  ->  (
i  e.  dom  f  <->  j  e.  dom  f ) )
2221anbi2d 684 . . . . 5  |-  ( i  =  j  ->  (
( f  e.  K  /\  i  e.  dom  f )  <->  ( f  e.  K  /\  j  e.  dom  f ) ) )
23 fveq2 5525 . . . . . 6  |-  ( i  =  j  ->  (
f `  i )  =  ( f `  j ) )
2423sseq1d 3205 . . . . 5  |-  ( i  =  j  ->  (
( f `  i
)  C_  B  <->  ( f `  j )  C_  B
) )
2522, 24imbi12d 311 . . . 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 3022 . . 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 8 . 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 262 1  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    C_ wss 3152   U_ciun 3905   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   ` cfv 5255    predc-bnj14 28086
This theorem is referenced by:  bnj1030  28390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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