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Theorem bnj1014 29308
Description: Technical lemma for bnj69 29356. 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 2432 . . . . . . . . 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 29280 . . . . . . . . . 10  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
65nfi 1541 . . . . . . . . 9  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
72, 6nfrex 2611 . . . . . . . 8  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
87nfab 2436 . . . . . . 7  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
91, 8nfcxfr 2429 . . . . . 6  |-  F/_ i B
109nfcri 2426 . . . . 5  |-  F/ i  g  e.  B
11 nfv 1609 . . . . 5  |-  F/ i  j  e.  dom  g
1210, 11nfan 1783 . . . 4  |-  F/ i ( g  e.  B  /\  j  e.  dom  g )
13 nfv 1609 . . . 4  |-  F/ i ( g `  j
)  C_  trCl ( X ,  A ,  R
)
1412, 13nfim 1781 . . 3  |-  F/ i ( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
1514nfri 1754 . 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 a9ev 1646 . . 3  |-  E. i 
i  =  j
171bnj1317 29170 . . . . . . . 8  |-  ( g  e.  B  ->  A. f 
g  e.  B )
1817nfi 1541 . . . . . . 7  |-  F/ f  g  e.  B
19 nfv 1609 . . . . . . 7  |-  F/ f  i  e.  dom  g
2018, 19nfan 1783 . . . . . 6  |-  F/ f ( g  e.  B  /\  i  e.  dom  g )
21 nfv 1609 . . . . . 6  |-  F/ f ( g `  i
)  C_  trCl ( X ,  A ,  R
)
2220, 21nfim 1781 . . . . 5  |-  F/ f ( ( g  e.  B  /\  i  e. 
dom  g )  -> 
( g `  i
)  C_  trCl ( X ,  A ,  R
) )
23 eleq1 2356 . . . . . . 7  |-  ( f  =  g  ->  (
f  e.  B  <->  g  e.  B ) )
24 dmeq 4895 . . . . . . . 8  |-  ( f  =  g  ->  dom  f  =  dom  g )
2524eleq2d 2363 . . . . . . 7  |-  ( f  =  g  ->  (
i  e.  dom  f  <->  i  e.  dom  g ) )
2623, 25anbi12d 691 . . . . . 6  |-  ( f  =  g  ->  (
( f  e.  B  /\  i  e.  dom  f )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
27 fveq1 5540 . . . . . . 7  |-  ( f  =  g  ->  (
f `  i )  =  ( g `  i ) )
2827sseq1d 3218 . . . . . 6  |-  ( f  =  g  ->  (
( f `  i
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
2926, 28imbi12d 311 . . . . 5  |-  ( 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 ) ) ) )
30 ssiun2 3961 . . . . . 6  |-  ( i  e.  dom  f  -> 
( f `  i
)  C_  U_ i  e. 
dom  f ( f `
 i ) )
31 ssiun2 3961 . . . . . . 7  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  U_ f  e.  B  U_ i  e.  dom  f ( f `  i ) )
32 bnj1014.13 . . . . . . . 8  |-  D  =  ( om  \  { (/)
} )
333, 4, 32, 1bnj882 29274 . . . . . . 7  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
3431, 33syl6sseqr 3238 . . . . . 6  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  trCl ( X ,  A ,  R ) )
3530, 34sylan9ssr 3206 . . . . 5  |-  ( ( f  e.  B  /\  i  e.  dom  f )  ->  ( f `  i )  C_  trCl ( X ,  A ,  R ) )
3622, 29, 35chvar 1939 . . . 4  |-  ( ( g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) )
37 eleq1 2356 . . . . . . 7  |-  ( j  =  i  ->  (
j  e.  dom  g  <->  i  e.  dom  g ) )
3837anbi2d 684 . . . . . 6  |-  ( j  =  i  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
39 fveq2 5541 . . . . . . 7  |-  ( j  =  i  ->  (
g `  j )  =  ( g `  i ) )
4039sseq1d 3218 . . . . . 6  |-  ( j  =  i  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
4138, 40imbi12d 311 . . . . 5  |-  ( 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 ) ) ) )
4241equcoms 1666 . . . 4  |-  ( 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 ) ) ) )
4336, 42mpbiri 224 . . 3  |-  ( i  =  j  ->  (
( g  e.  B  /\  j  e.  dom  g )  ->  (
g `  j )  C_ 
trCl ( X ,  A ,  R )
) )
4416, 43bnj101 29065 . 2  |-  E. i
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
4515, 44bnj1131 29135 1  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   U_ciun 3921   suc csuc 4410   omcom 4672   dom cdm 4705    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    trClc-bnj18 29035
This theorem is referenced by:  bnj1015  29309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-dm 4715  df-iota 5235  df-fv 5279  df-bnj18 29036
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