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Theorem bnj1110 29791
Description: Technical lemma for bnj69 29819. 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 29595 . . . . . . . 8  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
3 bnj219 29541 . . . . . . . . . . 11  |-  ( i  =  suc  j  -> 
j  _E  i )
43adantl 468 . . . . . . . . . 10  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  j  _E  i
)
54ancli 554 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  ( ( j  e.  n  /\  i  =  suc  j )  /\  j  _E  i )
)
6 df-3an 987 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  <->  ( ( j  e.  n  /\  i  =  suc  j )  /\  j  _E  i ) )
75, 6sylibr 216 . . . . . . . 8  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )
82, 7bnj1023 29592 . . . . . . 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 29615 . . . . . . . . . . 11  |-  ( ch 
->  n  e.  D
)
11103ad2ant3 1031 . . . . . . . . . 10  |-  ( ( th  /\  ta  /\  ch )  ->  n  e.  D )
12 bnj1110.19 . . . . . . . . . . 11  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
1312bnj1232 29615 . . . . . . . . . 10  |-  ( ph0  ->  i  e.  n )
1411, 13anim12ci 571 . . . . . . . . 9  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( i  e.  n  /\  n  e.  D
) )
1514anim2i 573 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =/=  (/)  /\  ( i  e.  n  /\  n  e.  D ) ) )
16 3anass 989 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D )  <->  ( i  =/=  (/)  /\  ( i  e.  n  /\  n  e.  D ) ) )
1715, 16sylibr 216 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
188, 17bnj1101 29596 . . . . . 6  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  j  _E  i
) )
19 3simpb 1006 . . . . . . . . 9  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  ( j  e.  n  /\  j  _E  i
) )
2012bnj1235 29616 . . . . . . . . . . 11  |-  ( ph0  ->  si )
2120ad2antll 735 . . . . . . . . . 10  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  si )
22 bnj1110.18 . . . . . . . . . 10  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
2321, 22sylib 200 . . . . . . . . 9  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  j  _E  i )  ->  et' ) )
2419, 23syl5 33 . . . . . . . 8  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( (
j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  et' ) )
2524a2i 14 . . . . . . 7  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  ->  et' ) )
26 pm3.43 873 . . . . . . 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 674 . . . . . 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 29529 . . . . 5  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) )
2912bnj1247 29620 . . . . . . 7  |-  ( ph0  ->  f  e.  K )
3029ad2antll 735 . . . . . 6  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  f  e.  K )
31 pm3.43i 458 . . . . . 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 5 . . . . 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 29529 . . . 4  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) )
34 fndm 5675 . . . . . . . . 9  |-  ( f  Fn  n  ->  dom  f  =  n )
359, 34bnj770 29574 . . . . . . . 8  |-  ( ch 
->  dom  f  =  n )
36353ad2ant3 1031 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch )  ->  dom  f  =  n )
3736ad2antrl 734 . . . . . 6  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  dom  f  =  n )
3837eleq2d 2514 . . . . 5  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  dom  f  <->  j  e.  n ) )
39 pm3.43i 458 . . . . 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 5 . . . 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 29529 . . 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 29514 . . . . . 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 29507 . . . . . 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 255 . . . . 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 314 . . . 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 1718 . . 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 213 . 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 1008 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  j  e.  n )
4948bnj706 29564 . . 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 1009 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  ->  i  =  suc  j
)
5150bnj706 29564 . . 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 29513 . . . . 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 466 . . . 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 29558 . . . . 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 236 . . . 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 29560 . . . . 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 200 . . . 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 685 . . 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 1188 . 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 29592 1  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622    \ cdif 3401    C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402    _E cep 4743   dom cdm 4834   suc csuc 5425    Fn wfn 5577   ` cfv 5582   omcom 6692    /\ w-bnj17 29491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-fn 5585  df-om 6693  df-bnj17 29492
This theorem is referenced by:  bnj1118  29793
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