Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1014 Structured version   Visualization version   Unicode version

Theorem bnj1014 29843
Description: Technical lemma for bnj69 29891. 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 2612 . . . . . . . . 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 29815 . . . . . . . . . 10  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  A. i
( f  Fn  n  /\  ph  /\  ps )
)
65nfi 1682 . . . . . . . . 9  |-  F/ i ( f  Fn  n  /\  ph  /\  ps )
72, 6nfrex 2848 . . . . . . . 8  |-  F/ i E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
87nfab 2616 . . . . . . 7  |-  F/_ i { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
91, 8nfcxfr 2610 . . . . . 6  |-  F/_ i B
109nfcri 2606 . . . . 5  |-  F/ i  g  e.  B
11 nfv 1769 . . . . 5  |-  F/ i  j  e.  dom  g
1210, 11nfan 2031 . . . 4  |-  F/ i ( g  e.  B  /\  j  e.  dom  g )
13 nfv 1769 . . . 4  |-  F/ i ( g `  j
)  C_  trCl ( X ,  A ,  R
)
1412, 13nfim 2023 . . 3  |-  F/ i ( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
1514nfri 1972 . 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 eleq1 2537 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  dom  g  <->  i  e.  dom  g ) )
1716anbi2d 718 . . . . 5  |-  ( j  =  i  ->  (
( g  e.  B  /\  j  e.  dom  g )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
18 fveq2 5879 . . . . . 6  |-  ( j  =  i  ->  (
g `  j )  =  ( g `  i ) )
1918sseq1d 3445 . . . . 5  |-  ( j  =  i  ->  (
( g `  j
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
2017, 19imbi12d 327 . . . 4  |-  ( 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 ) ) ) )
2120equcoms 1872 . . 3  |-  ( 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 ) ) ) )
221bnj1317 29705 . . . . . . 7  |-  ( g  e.  B  ->  A. f 
g  e.  B )
2322nfi 1682 . . . . . 6  |-  F/ f  g  e.  B
24 nfv 1769 . . . . . 6  |-  F/ f  i  e.  dom  g
2523, 24nfan 2031 . . . . 5  |-  F/ f ( g  e.  B  /\  i  e.  dom  g )
26 nfv 1769 . . . . 5  |-  F/ f ( g `  i
)  C_  trCl ( X ,  A ,  R
)
2725, 26nfim 2023 . . . 4  |-  F/ f ( ( g  e.  B  /\  i  e. 
dom  g )  -> 
( g `  i
)  C_  trCl ( X ,  A ,  R
) )
28 eleq1 2537 . . . . . 6  |-  ( f  =  g  ->  (
f  e.  B  <->  g  e.  B ) )
29 dmeq 5040 . . . . . . 7  |-  ( f  =  g  ->  dom  f  =  dom  g )
3029eleq2d 2534 . . . . . 6  |-  ( f  =  g  ->  (
i  e.  dom  f  <->  i  e.  dom  g ) )
3128, 30anbi12d 725 . . . . 5  |-  ( f  =  g  ->  (
( f  e.  B  /\  i  e.  dom  f )  <->  ( g  e.  B  /\  i  e.  dom  g ) ) )
32 fveq1 5878 . . . . . 6  |-  ( f  =  g  ->  (
f `  i )  =  ( g `  i ) )
3332sseq1d 3445 . . . . 5  |-  ( f  =  g  ->  (
( f `  i
)  C_  trCl ( X ,  A ,  R
)  <->  ( g `  i )  C_  trCl ( X ,  A ,  R ) ) )
3431, 33imbi12d 327 . . . 4  |-  ( 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 ) ) ) )
35 ssiun2 4312 . . . . 5  |-  ( i  e.  dom  f  -> 
( f `  i
)  C_  U_ i  e. 
dom  f ( f `
 i ) )
36 ssiun2 4312 . . . . . 6  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  U_ f  e.  B  U_ i  e.  dom  f ( f `  i ) )
37 bnj1014.13 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
383, 4, 37, 1bnj882 29809 . . . . . 6  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
3936, 38syl6sseqr 3465 . . . . 5  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  trCl ( X ,  A ,  R ) )
4035, 39sylan9ssr 3432 . . . 4  |-  ( ( f  e.  B  /\  i  e.  dom  f )  ->  ( f `  i )  C_  trCl ( X ,  A ,  R ) )
4127, 34, 40chvar 2119 . . 3  |-  ( ( g  e.  B  /\  i  e.  dom  g )  ->  ( g `  i )  C_  trCl ( X ,  A ,  R ) )
4221, 41spei 2118 . 2  |-  E. i
( ( g  e.  B  /\  j  e. 
dom  g )  -> 
( g `  j
)  C_  trCl ( X ,  A ,  R
) )
4315, 42bnj1131 29671 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   U_ciun 4269   dom cdm 4839   suc csuc 5432    Fn wfn 5584   ` cfv 5589   omcom 6711    predc-bnj14 29565    trClc-bnj18 29571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-dm 4849  df-iota 5553  df-fv 5597  df-bnj18 29572
This theorem is referenced by:  bnj1015  29844
  Copyright terms: Public domain W3C validator