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Theorem bnj1110 28682
Description: Technical lemma for bnj69 28710. 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 28485 . . . . . . . 8  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
3 bnj219 28431 . . . . . . . . . . 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 28482 . . . . . . 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 28506 . . . . . . . . . . 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 28506 . . . . . . . . . 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 28486 . . . . . 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 28507 . . . . . . . . . . 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 28419 . . . . 5  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) )
2912bnj1247 28511 . . . . . . 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 28419 . . . 4  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f  e.  K  /\  ( ( j  e.  n  /\  i  =  suc  j  /\  j  _E  i )  /\  et' ) ) )
34 fndm 5477 . . . . . . . . 9  |-  ( f  Fn  n  ->  dom  f  =  n )
359, 34bnj770 28463 . . . . . . . 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 2447 . . . . 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 28419 . . 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 28404 . . . . . 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 28397 . . . . . 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 1589 . . 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 28453 . . 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 28453 . . 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 28403 . . . . 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 28447 . . . . 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 28449 . . . . 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 28482 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 1547    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253    C_ wss 3256   (/)c0 3564   {csn 3750   class class class wbr 4146    _E cep 4426   suc csuc 4517   omcom 4778   dom cdm 4811    Fn wfn 5382   ` cfv 5387    /\ w-bnj17 28381
This theorem is referenced by:  bnj1118  28684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-fn 5390  df-bnj17 28382
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