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Theorem bnj1014 33503
Description: Technical lemma for bnj69 33551. 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
bnj1014.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1014.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1014.13  |-  D  =  ( om  \  { (/)
} )
bnj1014.14  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj1014  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
Distinct variable groups:    A, f,
i, n, y    D, i    R, f, i, n, y    f, X, i, n, y    f, g, i    i, j    ph, i
Allowed substitution hints:    ph( y, f, g, j, n)    ps( y, f, g, i, j, n)    A( g, j)    B( y, f, g, i, j, n)    D( y, f, g, j, n)    R( g,
j)    X( g, j)

Proof of Theorem bnj1014
StepHypRef Expression
1 bnj1014.14 . . . . . . 7  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
2 nfcv 2629 . . . . . . . . 9  |-  F/_ i D
3 bnj1014.1 . . . . . . . . . . 11  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
4 bnj1014.2 . . . . . . . . . . 11  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
53, 4bnj911 33475 . . . . . . . . . 10  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
65nfi 1606 . . . . . . . . 9  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
72, 6nfrex 2930 . . . . . . . 8  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
87nfab 2633 . . . . . . 7  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
91, 8nfcxfr 2627 . . . . . 6  |-  F/_ i B
109nfcri 2622 . . . . 5  |-  F/ i  g  e.  B
11 nfv 1683 . . . . 5  |-  F/ i  j  e.  dom  g
1210, 11nfan 1875 . . . 4  |-  F/ i ( g  e.  B  /\  j  e.  dom  g )
13 nfv 1683 . . . 4  |-  F/ i ( g `  j
)  C_  trCl ( X ,  A ,  R
)
1412, 13nfim 1867 . . 3  |-  F/ i ( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
1514nfri 1822 . 2  |-  ( ( ( g  e.  B  /\  j  e.  dom  g )  ->  (
g `  j )  C_ 
trCl ( X ,  A ,  R )
)  ->  A. i
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) ) )
16 eleq1 2539 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  dom  g  <->  i  e.  dom  g ) )
1716anbi2d 703 . . . . 5  |-  ( j  =  i  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
18 fveq2 5872 . . . . . 6  |-  ( j  =  i  ->  (
g `  j )  =  ( g `  i ) )
1918sseq1d 3536 . . . . 5  |-  ( j  =  i  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
2017, 19imbi12d 320 . . . 4  |-  ( j  =  i  ->  (
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )  <->  ( (
g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) ) )
2120equcoms 1744 . . 3  |-  ( i  =  j  ->  (
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )  <->  ( (
g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) ) )
221bnj1317 33365 . . . . . . 7  |-  ( g  e.  B  ->  A. f 
g  e.  B )
2322nfi 1606 . . . . . 6  |-  F/ f  g  e.  B
24 nfv 1683 . . . . . 6  |-  F/ f  i  e.  dom  g
2523, 24nfan 1875 . . . . 5  |-  F/ f ( g  e.  B  /\  i  e.  dom  g )
26 nfv 1683 . . . . 5  |-  F/ f ( g `  i
)  C_  trCl ( X ,  A ,  R
)
2725, 26nfim 1867 . . . 4  |-  F/ f ( ( g  e.  B  /\  i  e. 
dom  g )  -> 
( g `  i
)  C_  trCl ( X ,  A ,  R
) )
28 eleq1 2539 . . . . . 6  |-  ( f  =  g  ->  (
f  e.  B  <->  g  e.  B ) )
29 dmeq 5209 . . . . . . 7  |-  ( f  =  g  ->  dom  f  =  dom  g )
3029eleq2d 2537 . . . . . 6  |-  ( f  =  g  ->  (
i  e.  dom  f  <->  i  e.  dom  g ) )
3128, 30anbi12d 710 . . . . 5  |-  ( f  =  g  ->  (
( f  e.  B  /\  i  e.  dom  f )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
32 fveq1 5871 . . . . . 6  |-  ( f  =  g  ->  (
f `  i )  =  ( g `  i ) )
3332sseq1d 3536 . . . . 5  |-  ( f  =  g  ->  (
( f `  i
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
3431, 33imbi12d 320 . . . 4  |-  ( f  =  g  ->  (
( ( f  e.  B  /\  i  e. 
dom  f )  -> 
( f `  i
)  C_  trCl ( X ,  A ,  R
) )  <->  ( (
g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) ) )
35 ssiun2 4374 . . . . 5  |-  ( i  e.  dom  f  -> 
( f `  i
)  C_  U_ i  e. 
dom  f ( f `
 i ) )
36 ssiun2 4374 . . . . . 6  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  U_ f  e.  B  U_ i  e.  dom  f ( f `  i ) )
37 bnj1014.13 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
383, 4, 37, 1bnj882 33469 . . . . . 6  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
3936, 38syl6sseqr 3556 . . . . 5  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  trCl ( X ,  A ,  R ) )
4035, 39sylan9ssr 3523 . . . 4  |-  ( ( f  e.  B  /\  i  e.  dom  f )  ->  ( f `  i )  C_  trCl ( X ,  A ,  R ) )
4127, 34, 40chvar 1982 . . 3  |-  ( ( g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) )
4221, 41spei 1981 . 2  |-  E. i
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
4315, 42bnj1131 33331 1  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818    \ cdif 3478    C_ wss 3481   (/)c0 3790   {csn 4033   U_ciun 4331   suc csuc 4886   dom cdm 5005    Fn wfn 5589   ` cfv 5594   omcom 6695    predc-bnj14 33226    trClc-bnj18 33232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602  df-bnj18 33233
This theorem is referenced by:  bnj1015  33504
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