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Theorem bnj1015 29269
Description: Technical lemma for bnj69 29316. 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 2957 . 2  |-  J  e. 
_V
3 eleq1 2495 . . . 4  |-  ( j  =  J  ->  (
j  e.  dom  G  <->  J  e.  dom  G ) )
43anbi2d 685 . . 3  |-  ( j  =  J  ->  (
( G  e.  B  /\  j  e.  dom  G )  <->  ( G  e.  B  /\  J  e. 
dom  G ) ) )
5 fveq2 5720 . . . 4  |-  ( j  =  J  ->  ( G `  j )  =  ( G `  J ) )
65sseq1d 3367 . . 3  |-  ( j  =  J  ->  (
( G `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( G `  J )  C_  trCl ( X ,  A ,  R ) ) )
74, 6imbi12d 312 . 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 2957 . . 3  |-  G  e. 
_V
10 eleq1 2495 . . . . 5  |-  ( g  =  G  ->  (
g  e.  B  <->  G  e.  B ) )
11 dmeq 5062 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
1211eleq2d 2502 . . . . 5  |-  ( g  =  G  ->  (
j  e.  dom  g  <->  j  e.  dom  G ) )
1310, 12anbi12d 692 . . . 4  |-  ( g  =  G  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( G  e.  B  /\  j  e.  dom  G ) ) )
14 fveq1 5719 . . . . 5  |-  ( g  =  G  ->  (
g `  j )  =  ( G `  j ) )
1514sseq1d 3367 . . . 4  |-  ( g  =  G  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( G `  j )  C_  trCl ( X ,  A ,  R ) ) )
1613, 15imbi12d 312 . . 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 29268 . . 3  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
229, 16, 21vtocl 2998 . 2  |-  ( ( G  e.  B  /\  j  e.  dom  G )  ->  ( G `  j )  C_  trCl ( X ,  A ,  R ) )
232, 7, 22vtocl 2998 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   U_ciun 4085   suc csuc 4575   omcom 4837   dom cdm 4870    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    trClc-bnj18 28995
This theorem is referenced by:  bnj1018  29270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454  df-bnj18 28996
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