Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1015 Structured version   Unicode version

Theorem bnj1015 31959
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
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 2987 . 2  |-  J  e. 
_V
3 eleq1 2503 . . . 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 5696 . . . 4  |-  ( j  =  J  ->  ( G `  j )  =  ( G `  J ) )
65sseq1d 3388 . . 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 2987 . . 3  |-  G  e. 
_V
10 eleq1 2503 . . . . 5  |-  ( g  =  G  ->  (
g  e.  B  <->  G  e.  B ) )
11 dmeq 5045 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
1211eleq2d 2510 . . . . 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 5695 . . . . 5  |-  ( g  =  G  ->  (
g `  j )  =  ( G `  j ) )
1514sseq1d 3388 . . . 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 31958 . . 3  |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j )  C_  trCl ( X ,  A ,  R ) )
229, 16, 21vtocl 3029 . 2  |-  ( ( G  e.  B  /\  j  e.  dom  G )  ->  ( G `  j )  C_  trCl ( X ,  A ,  R ) )
232, 7, 22vtocl 3029 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:  bnj1018  31960
  Copyright terms: Public domain W3C validator