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Theorem bnj1110 29288
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
bnj1110.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1110.7  |-  D  =  ( om  \  { (/)
} )
bnj1110.18  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
bnj1110.19  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
bnj1110.26  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
Assertion
Ref Expression
bnj1110  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
Distinct variable groups:    D, j    i, j    j, n
Allowed substitution hints:    ph( f, i, j, n)    ps( f,
i, j, n)    ch( f, i, j, n)    th( f,
i, j, n)    ta( f, i, j, n)    si( f,
i, j, n)    B( f, i, j, n)    D( f, i, n)    K( f,
i, j, n)    et'( f, i, j, n)    ph0( f, i, j, n)

Proof of Theorem bnj1110
StepHypRef Expression
1 bnj1110.7 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
21bnj1098 29091 . . . . . . . 8  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
3 bnj219 29037 . . . . . . . . . . 11  |-  ( i  =  suc  j  -> 
j  _E  i )
43adantl 453 . . . . . . . . . 10  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  j  _E  i
)
54ancli 535 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  ( ( j  e.  n  /\  i  =  suc  j )  /\  j  _E  i )
)
6 df-3an 938 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  <->  ( ( j  e.  n  /\  i  =  suc  j )  /\  j  _E  i ) )
75, 6sylibr 204 . . . . . . . 8  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )
82, 7bnj1023 29088 . . . . . . 7  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )
9 bnj1110.3 . . . . . . . . . . . 12  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
109bnj1232 29112 . . . . . . . . . . 11  |-  ( ch 
->  n  e.  D
)
11103ad2ant3 980 . . . . . . . . . 10  |-  ( ( th  /\  ta  /\  ch )  ->  n  e.  D )
12 bnj1110.19 . . . . . . . . . . 11  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
1312bnj1232 29112 . . . . . . . . . 10  |-  ( ph0  ->  i  e.  n )
1411, 13anim12ci 551 . . . . . . . . 9  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( i  e.  n  /\  n  e.  D
) )
1514anim2i 553 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =/=  (/)  /\  ( i  e.  n  /\  n  e.  D ) ) )
16 3anass 940 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( i  =/=  (/)  /\  ( i  e.  n  /\  n  e.  D ) ) )
1715, 16sylibr 204 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
188, 17bnj1101 29092 . . . . . 6  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
) )
19 3simpb 955 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  ( j  e.  n  /\  j  _E  i
) )
2012bnj1235 29113 . . . . . . . . . . 11  |-  ( ph0  ->  si )
2120ad2antll 710 . . . . . . . . . 10  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  si )
22 bnj1110.18 . . . . . . . . . 10  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
2321, 22sylib 189 . . . . . . . . 9  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  j  _E  i )  ->  et' ) )
2419, 23syl5 30 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  et' ) )
2524a2i 13 . . . . . . 7  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  ->  et' ) )
26 pm3.43 833 . . . . . . 7  |-  ( ( ( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
) )  /\  (
( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  et' ) )  ->  ( ( i  =/=  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) )
2725, 26mpdan 650 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) )
2818, 27bnj101 29025 . . . . 5  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) )
2912bnj1247 29117 . . . . . . 7  |-  ( ph0  ->  f  e.  K )
3029ad2antll 710 . . . . . 6  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  f  e.  K )
31 pm3.43i 443 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  f  e.  K )  ->  (
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) )  ->  ( ( i  =/=  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 )
)  ->  ( f  e.  K  /\  (
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' ) ) ) ) )
3230, 31ax-mp 8 . . . . 5  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) )  ->  (
( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f  e.  K  /\  (
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' ) ) ) )
3328, 32bnj101 29025 . . . 4  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) )
34 fndm 5536 . . . . . . . . 9  |-  ( f  Fn  n  ->  dom  f  =  n )
359, 34bnj770 29069 . . . . . . . 8  |-  ( ch 
->  dom  f  =  n )
36353ad2ant3 980 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch )  ->  dom  f  =  n )
3736ad2antrl 709 . . . . . 6  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  dom  f  =  n )
3837eleq2d 2502 . . . . 5  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  dom  f  <->  j  e.  n ) )
39 pm3.43i 443 . . . . 5  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  dom  f  <->  j  e.  n ) )  -> 
( ( ( i  =/=  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 )
)  ->  ( f  e.  K  /\  (
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' ) ) )  ->  ( (
i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) ) ) )
4038, 39ax-mp 8 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f  e.  K  /\  (
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' ) ) )  ->  ( (
i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) ) )
4133, 40bnj101 29025 . . 3  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e. 
dom  f  <->  j  e.  n )  /\  (
f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) )
42 bnj268 29010 . . . . . 6  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  f  e.  K  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' )  <->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  f  e.  K  /\  et' ) )
43 bnj251 29003 . . . . . 6  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  f  e.  K  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  et' )  <->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) )
4442, 43bitr3i 243 . . . . 5  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  <->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) )
4544imbi2i 304 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  f  e.  K  /\  et' ) )  <->  ( (
i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) ) )
4645exbii 1592 . . 3  |-  ( E. j ( ( i  =/=  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 )
)  ->  ( (
j  e.  dom  f  <->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
)  /\  f  e.  K  /\  et' ) )  <->  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e. 
dom  f  <->  j  e.  n )  /\  (
f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) ) ) )
4741, 46mpbir 201 . 2  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e. 
dom  f  <->  j  e.  n )  /\  (
j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' ) )
48 simp1 957 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  j  e.  n )
4948bnj706 29059 . . 3  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
j  e.  n )
50 simp2 958 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  i  =  suc  j
)
5150bnj706 29059 . . 3  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
i  =  suc  j
)
52 bnj258 29009 . . . . 5  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  <->  ( (
( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' )  /\  f  e.  K )
)
5352simprbi 451 . . . 4  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
f  e.  K )
54 bnj642 29053 . . . . 5  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
( j  e.  dom  f 
<->  j  e.  n ) )
5549, 54mpbird 224 . . . 4  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
j  e.  dom  f
)
56 bnj645 29055 . . . . 5  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  ->  et' )
57 bnj1110.26 . . . . 5  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
5856, 57sylib 189 . . . 4  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
( ( f  e.  K  /\  j  e. 
dom  f )  -> 
( f `  j
)  C_  B )
)
5953, 55, 58mp2and 661 . . 3  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
( f `  j
)  C_  B )
6049, 51, 593jca 1134 . 2  |-  ( ( ( j  e.  dom  f 
<->  j  e.  n )  /\  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  f  e.  K  /\  et' )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
6147, 60bnj1023 29088 1  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    _E cep 4484   suc csuc 4575   omcom 4837   dom cdm 4870    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987
This theorem is referenced by:  bnj1118  29290
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-fn 5449  df-bnj17 28988
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