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Theorem bnj1118 34183
Description: Technical lemma for bnj69 34209. 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
bnj1118.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1118.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1118.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1118.7  |-  D  =  ( om  \  { (/)
} )
bnj1118.18  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
bnj1118.19  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
bnj1118.26  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
Assertion
Ref Expression
bnj1118  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f `  i
)  C_  B )
Distinct variable groups:    A, i,
j, y    y, B    D, j    R, i, j, y   
f, i, j, y   
i, n, j
Allowed substitution hints:    ph( y, f, i, j, n)    ps( y, f, i, j, n)    ch( y, f, i, j, n)    th( y, f, i, j, n)    ta( y,
f, i, j, n)    si( y, f, i, j, n)    A( f, n)    B( f, i, j, n)    D( y, f, i, n)    R( f, n)    K( y, f, i, j, n)    X( y, f, i, j, n)    et'( y, f, i, j, n)    ph0( y, f, i, j, n)

Proof of Theorem bnj1118
StepHypRef Expression
1 bnj1118.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj1118.7 . . . 4  |-  D  =  ( om  \  { (/)
} )
3 bnj1118.18 . . . 4  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
4 bnj1118.19 . . . 4  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
5 bnj1118.26 . . . 4  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
61, 2, 3, 4, 5bnj1110 34181 . . 3  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
7 ancl 546 . . 3  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  /\  ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) ) ) )
86, 7bnj101 33919 . 2  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  /\  ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) ) )
9 simpr2 1003 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
i  =  suc  j
)
101bnj1254 34011 . . . . . . 7  |-  ( ch 
->  ps )
11103ad2ant3 1019 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ps )
1211ad2antrl 727 . . . . 5  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ps )
1312adantr 465 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  ->  ps )
141bnj1232 34005 . . . . . . . . 9  |-  ( ch 
->  n  e.  D
)
15143ad2ant3 1019 . . . . . . . 8  |-  ( ( th  /\  ta  /\  ch )  ->  n  e.  D )
1615ad2antrl 727 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  n  e.  D )
1716adantr 465 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  ->  n  e.  D )
18 simpr1 1002 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
j  e.  n )
192bnj923 33969 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
2019anim1i 568 . . . . . . 7  |-  ( ( n  e.  D  /\  j  e.  n )  ->  ( n  e.  om  /\  j  e.  n ) )
2120ancomd 451 . . . . . 6  |-  ( ( n  e.  D  /\  j  e.  n )  ->  ( j  e.  n  /\  n  e.  om ) )
2217, 18, 21syl2anc 661 . . . . 5  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( j  e.  n  /\  n  e.  om ) )
23 elnn 6709 . . . . 5  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
2422, 23syl 16 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
j  e.  om )
254bnj1232 34005 . . . . . 6  |-  ( ph0  ->  i  e.  n )
2625adantl 466 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  i  e.  n )
2726ad2antlr 726 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
i  e.  n )
289, 13, 24, 27bnj951 33977 . . 3  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( i  =  suc  j  /\  ps  /\  j  e.  om  /\  i  e.  n ) )
29 bnj1118.5 . . . . . . 7  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
3029simp2bi 1012 . . . . . 6  |-  ( ta 
->  TrFo ( B ,  A ,  R )
)
31303ad2ant2 1018 . . . . 5  |-  ( ( th  /\  ta  /\  ch )  ->  TrFo ( B ,  A ,  R ) )
3231ad2antrl 727 . . . 4  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  TrFo ( B ,  A ,  R
) )
33 simp3 998 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j
)  C_  B )  ->  ( f `  j
)  C_  B )
3432, 33anim12i 566 . . 3  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
(  TrFo ( B ,  A ,  R )  /\  ( f `  j
)  C_  B )
)
35 bnj256 33901 . . . . 5  |-  ( ( i  =  suc  j  /\  ps  /\  j  e. 
om  /\  i  e.  n )  <->  ( (
i  =  suc  j  /\  ps )  /\  (
j  e.  om  /\  i  e.  n )
) )
36 bnj1118.2 . . . . . . . . . 10  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3736bnj1112 34182 . . . . . . . . 9  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
3837biimpi 194 . . . . . . . 8  |-  ( ps 
->  A. j ( ( j  e.  om  /\  suc  j  e.  n
)  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
393819.21bi 1870 . . . . . . 7  |-  ( ps 
->  ( ( j  e. 
om  /\  suc  j  e.  n )  ->  (
f `  suc  j )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) )
40 eleq1 2529 . . . . . . . . 9  |-  ( i  =  suc  j  -> 
( i  e.  n  <->  suc  j  e.  n ) )
4140anbi2d 703 . . . . . . . 8  |-  ( i  =  suc  j  -> 
( ( j  e. 
om  /\  i  e.  n )  <->  ( j  e.  om  /\  suc  j  e.  n ) ) )
42 fveq2 5872 . . . . . . . . 9  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
4342eqeq1d 2459 . . . . . . . 8  |-  ( i  =  suc  j  -> 
( ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R )  <->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
4441, 43imbi12d 320 . . . . . . 7  |-  ( i  =  suc  j  -> 
( ( ( j  e.  om  /\  i  e.  n )  ->  (
f `  i )  =  U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) )  <->  ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
4539, 44syl5ibr 221 . . . . . 6  |-  ( i  =  suc  j  -> 
( ps  ->  (
( j  e.  om  /\  i  e.  n )  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
4645imp31 432 . . . . 5  |-  ( ( ( i  =  suc  j  /\  ps )  /\  ( j  e.  om  /\  i  e.  n ) )  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
4735, 46sylbi 195 . . . 4  |-  ( ( i  =  suc  j  /\  ps  /\  j  e. 
om  /\  i  e.  n )  ->  (
f `  i )  =  U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) )
48 df-bnj19 33892 . . . . . . 7  |-  (  TrFo ( B ,  A ,  R )  <->  A. y  e.  B  pred ( y ,  A ,  R
)  C_  B )
49 ssralv 3560 . . . . . . 7  |-  ( ( f `  j ) 
C_  B  ->  ( A. y  e.  B  pred ( y ,  A ,  R )  C_  B  ->  A. y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B ) )
5048, 49syl5bi 217 . . . . . 6  |-  ( ( f `  j ) 
C_  B  ->  (  TrFo ( B ,  A ,  R )  ->  A. y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  B ) )
5150impcom 430 . . . . 5  |-  ( ( 
TrFo ( B ,  A ,  R )  /\  ( f `  j
)  C_  B )  ->  A. y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B )
52 iunss 4373 . . . . 5  |-  ( U_ y  e.  ( f `  j )  pred (
y ,  A ,  R )  C_  B  <->  A. y  e.  ( f `
 j )  pred ( y ,  A ,  R )  C_  B
)
5351, 52sylibr 212 . . . 4  |-  ( ( 
TrFo ( B ,  A ,  R )  /\  ( f `  j
)  C_  B )  ->  U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B )
54 sseq1 3520 . . . . 5  |-  ( ( f `  i )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R )  ->  ( ( f `
 i )  C_  B 
<-> 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B ) )
5554biimpar 485 . . . 4  |-  ( ( ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R )  /\  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  B )  -> 
( f `  i
)  C_  B )
5647, 53, 55syl2an 477 . . 3  |-  ( ( ( i  =  suc  j  /\  ps  /\  j  e.  om  /\  i  e.  n )  /\  (  TrFo ( B ,  A ,  R )  /\  (
f `  j )  C_  B ) )  -> 
( f `  i
)  C_  B )
5728, 34, 56syl2anc 661 . 2  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( f `  i
)  C_  B )
588, 57bnj1023 33982 1  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f `  i
)  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109    \ cdif 3468    C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332   class class class wbr 4456    _E cep 4798   suc csuc 4889   dom cdm 5008    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33881    predc-bnj14 33883    TrFow-bnj19 33891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-iota 5557  df-fn 5597  df-fv 5602  df-om 6700  df-bnj17 33882  df-bnj19 33892
This theorem is referenced by:  bnj1030  34186
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