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Theorem bnj1015 34120
Description: Technical lemma for bnj69 34167. 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
bnj1015.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1015.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1015.13  |-  D  =  ( om  \  { (/)
} )
bnj1015.14  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1015.15  |-  G  e.  V
bnj1015.16  |-  J  e.  V
Assertion
Ref Expression
bnj1015  |-  ( ( 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    ph, i
Allowed substitution hints:    ph( y, f, n)    ps( y, f, i, n)    B( y, f, i, n)    D( y, f, n)    G( y, f, i, n)    J( y, f, i, n)    V( y, f, i, n)

Proof of Theorem bnj1015
Dummy variables  g 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1015.16 . . 3  |-  J  e.  V
21elexi 3119 . 2  |-  J  e. 
_V
3 eleq1 2529 . . . 4  |-  ( j  =  J  ->  (
j  e.  dom  G  <->  J  e.  dom  G ) )
43anbi2d 703 . . 3  |-  ( j  =  J  ->  (
( G  e.  B  /\  j  e.  dom  G )  <->  ( G  e.  B  /\  J  e. 
dom  G ) ) )
5 fveq2 5872 . . . 4  |-  ( j  =  J  ->  ( G `  j )  =  ( G `  J ) )
65sseq1d 3526 . . 3  |-  ( j  =  J  ->  (
( G `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( G `  J )  C_  trCl ( X ,  A ,  R ) ) )
74, 6imbi12d 320 . 2  |-  ( j  =  J  ->  (
( ( G  e.  B  /\  j  e. 
dom  G )  -> 
( G `  j
)  C_  trCl ( X ,  A ,  R
) )  <->  ( ( G  e.  B  /\  J  e.  dom  G )  ->  ( G `  J )  C_  trCl ( X ,  A ,  R ) ) ) )
8 bnj1015.15 . . . 4  |-  G  e.  V
98elexi 3119 . . 3  |-  G  e. 
_V
10 eleq1 2529 . . . . 5  |-  ( g  =  G  ->  (
g  e.  B  <->  G  e.  B ) )
11 dmeq 5213 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
1211eleq2d 2527 . . . . 5  |-  ( g  =  G  ->  (
j  e.  dom  g  <->  j  e.  dom  G ) )
1310, 12anbi12d 710 . . . 4  |-  ( g  =  G  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( G  e.  B  /\  j  e.  dom  G ) ) )
14 fveq1 5871 . . . . 5  |-  ( g  =  G  ->  (
g `  j )  =  ( G `  j ) )
1514sseq1d 3526 . . . 4  |-  ( g  =  G  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( G `  j )  C_  trCl ( X ,  A ,  R ) ) )
1613, 15imbi12d 320 . . 3  |-  ( g  =  G  ->  (
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )  <->  ( ( G  e.  B  /\  j  e.  dom  G )  ->  ( G `  j )  C_  trCl ( X ,  A ,  R ) ) ) )
17 bnj1015.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
18 bnj1015.2 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
19 bnj1015.13 . . . 4  |-  D  =  ( om  \  { (/)
} )
20 bnj1015.14 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
2117, 18, 19, 20bnj1014 34119 . . 3  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
229, 16, 21vtocl 3161 . 2  |-  ( ( G  e.  B  /\  j  e.  dom  G )  ->  ( G `  j )  C_  trCl ( X ,  A ,  R ) )
232, 7, 22vtocl 3161 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 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808    \ cdif 3468    C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332   suc csuc 4889   dom cdm 5008    Fn wfn 5589   ` cfv 5594   omcom 6699    predc-bnj14 33841    trClc-bnj18 33847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602  df-bnj18 33848
This theorem is referenced by:  bnj1018  34121
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