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Theorem bnj1014 31958
Description: Technical lemma for bnj69 32006. 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 2584 . . . . . . . . 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 31930 . . . . . . . . . 10  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
65nfi 1596 . . . . . . . . 9  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
72, 6nfrex 2776 . . . . . . . 8  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
87nfab 2588 . . . . . . 7  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
91, 8nfcxfr 2581 . . . . . 6  |-  F/_ i B
109nfcri 2578 . . . . 5  |-  F/ i  g  e.  B
11 nfv 1673 . . . . 5  |-  F/ i  j  e.  dom  g
1210, 11nfan 1861 . . . 4  |-  F/ i ( g  e.  B  /\  j  e.  dom  g )
13 nfv 1673 . . . 4  |-  F/ i ( g `  j
)  C_  trCl ( X ,  A ,  R
)
1412, 13nfim 1853 . . 3  |-  F/ i ( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
1514nfri 1808 . 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 ax6ev 1710 . . 3  |-  E. i 
i  =  j
171bnj1317 31820 . . . . . . . 8  |-  ( g  e.  B  ->  A. f 
g  e.  B )
1817nfi 1596 . . . . . . 7  |-  F/ f  g  e.  B
19 nfv 1673 . . . . . . 7  |-  F/ f  i  e.  dom  g
2018, 19nfan 1861 . . . . . 6  |-  F/ f ( g  e.  B  /\  i  e.  dom  g )
21 nfv 1673 . . . . . 6  |-  F/ f ( g `  i
)  C_  trCl ( X ,  A ,  R
)
2220, 21nfim 1853 . . . . 5  |-  F/ f ( ( g  e.  B  /\  i  e. 
dom  g )  -> 
( g `  i
)  C_  trCl ( X ,  A ,  R
) )
23 eleq1 2503 . . . . . . 7  |-  ( f  =  g  ->  (
f  e.  B  <->  g  e.  B ) )
24 dmeq 5045 . . . . . . . 8  |-  ( f  =  g  ->  dom  f  =  dom  g )
2524eleq2d 2510 . . . . . . 7  |-  ( f  =  g  ->  (
i  e.  dom  f  <->  i  e.  dom  g ) )
2623, 25anbi12d 710 . . . . . 6  |-  ( f  =  g  ->  (
( f  e.  B  /\  i  e.  dom  f )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
27 fveq1 5695 . . . . . . 7  |-  ( f  =  g  ->  (
f `  i )  =  ( g `  i ) )
2827sseq1d 3388 . . . . . 6  |-  ( f  =  g  ->  (
( f `  i
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
2926, 28imbi12d 320 . . . . 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 4218 . . . . . 6  |-  ( i  e.  dom  f  -> 
( f `  i
)  C_  U_ i  e. 
dom  f ( f `
 i ) )
31 ssiun2 4218 . . . . . . 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 31924 . . . . . . 7  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
3431, 33syl6sseqr 3408 . . . . . 6  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  trCl ( X ,  A ,  R ) )
3530, 34sylan9ssr 3375 . . . . 5  |-  ( ( f  e.  B  /\  i  e.  dom  f )  ->  ( f `  i )  C_  trCl ( X ,  A ,  R ) )
3622, 29, 35chvar 1957 . . . 4  |-  ( ( g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) )
37 eleq1 2503 . . . . . . 7  |-  ( j  =  i  ->  (
j  e.  dom  g  <->  i  e.  dom  g ) )
3837anbi2d 703 . . . . . 6  |-  ( j  =  i  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
39 fveq2 5696 . . . . . . 7  |-  ( j  =  i  ->  (
g `  j )  =  ( g `  i ) )
4039sseq1d 3388 . . . . . 6  |-  ( j  =  i  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
4138, 40imbi12d 320 . . . . 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 1733 . . . 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 233 . . 3  |-  ( i  =  j  ->  (
( g  e.  B  /\  j  e.  dom  g )  ->  (
g `  j )  C_ 
trCl ( X ,  A ,  R )
) )
4416, 43bnj101 31717 . 2  |-  E. i
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
4515, 44bnj1131 31786 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 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721    \ cdif 3330    C_ wss 3333   (/)c0 3642   {csn 3882   U_ciun 4176   suc csuc 4726   dom cdm 4845    Fn wfn 5418   ` cfv 5423   omcom 6481    predc-bnj14 31681    trClc-bnj18 31687
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 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-dm 4855  df-iota 5386  df-fv 5431  df-bnj18 31688
This theorem is referenced by:  bnj1015  31959
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