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Theorem bnj1118 28691
Description: Technical lemma for bnj69 28717. 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 28689 . . 3  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B
) )
7 ancl 530 . . 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 28426 . 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 964 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
i  =  suc  j
)
101bnj1254 28519 . . . . . . 7  |-  ( ch 
->  ps )
11103ad2ant3 980 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ps )
1211ad2antrl 709 . . . . 5  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ps )
1312adantr 452 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  ->  ps )
141bnj1232 28513 . . . . . . . . 9  |-  ( ch 
->  n  e.  D
)
15143ad2ant3 980 . . . . . . . 8  |-  ( ( th  /\  ta  /\  ch )  ->  n  e.  D )
1615ad2antrl 709 . . . . . . 7  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  n  e.  D )
1716adantr 452 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  ->  n  e.  D )
18 simpr1 963 . . . . . 6  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
j  e.  n )
192bnj923 28475 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
2019anim1i 552 . . . . . . 7  |-  ( ( n  e.  D  /\  j  e.  n )  ->  ( n  e.  om  /\  j  e.  n ) )
2120ancomd 439 . . . . . 6  |-  ( ( n  e.  D  /\  j  e.  n )  ->  ( j  e.  n  /\  n  e.  om ) )
2217, 18, 21syl2anc 643 . . . . 5  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( j  e.  n  /\  n  e.  om ) )
23 elnn 4795 . . . . 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 28513 . . . . . 6  |-  ( ph0  ->  i  e.  n )
2625adantl 453 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  i  e.  n )
2726ad2antlr 708 . . . 4  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
i  e.  n )
289, 13, 24, 27bnj951 28484 . . 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 973 . . . . . 6  |-  ( ta 
->  TrFo ( B ,  A ,  R )
)
31303ad2ant2 979 . . . . 5  |-  ( ( th  /\  ta  /\  ch )  ->  TrFo ( B ,  A ,  R ) )
3231ad2antrl 709 . . . 4  |-  ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  TrFo ( B ,  A ,  R
) )
33 simp3 959 . . . 4  |-  ( ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j
)  C_  B )  ->  ( f `  j
)  C_  B )
3432, 33anim12i 550 . . 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 28408 . . . . 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 28690 . . . . . . . . 9  |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n )  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
3837biimpi 187 . . . . . . . 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 1766 . . . . . . 7  |-  ( ps 
->  ( ( j  e. 
om  /\  suc  j  e.  n )  ->  (
f `  suc  j )  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) )
40 eleq1 2447 . . . . . . . . 9  |-  ( i  =  suc  j  -> 
( i  e.  n  <->  suc  j  e.  n ) )
4140anbi2d 685 . . . . . . . 8  |-  ( i  =  suc  j  -> 
( ( j  e. 
om  /\  i  e.  n )  <->  ( j  e.  om  /\  suc  j  e.  n ) ) )
42 fveq2 5668 . . . . . . . . 9  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
4342eqeq1d 2395 . . . . . . . 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 312 . . . . . . 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 213 . . . . . 6  |-  ( i  =  suc  j  -> 
( ps  ->  (
( j  e.  om  /\  i  e.  n )  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
4645imp31 422 . . . . 5  |-  ( ( ( i  =  suc  j  /\  ps )  /\  ( j  e.  om  /\  i  e.  n ) )  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
4735, 46sylbi 188 . . . 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 28399 . . . . . . 7  |-  (  TrFo ( B ,  A ,  R )  <->  A. y  e.  B  pred ( y ,  A ,  R
)  C_  B )
49 ssralv 3350 . . . . . . 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 209 . . . . . 6  |-  ( ( f `  j ) 
C_  B  ->  (  TrFo ( B ,  A ,  R )  ->  A. y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  B ) )
5150impcom 420 . . . . 5  |-  ( ( 
TrFo ( B ,  A ,  R )  /\  ( f `  j
)  C_  B )  ->  A. y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B )
52 iunss 4073 . . . . 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 204 . . . 4  |-  ( ( 
TrFo ( B ,  A ,  R )  /\  ( f `  j
)  C_  B )  ->  U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) 
C_  B )
54 sseq1 3312 . . . . 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 472 . . . 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 464 . . 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 643 . 2  |-  ( ( ( i  =/=  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  /\  ( j  e.  n  /\  i  =  suc  j  /\  (
f `  j )  C_  B ) )  -> 
( f `  i
)  C_  B )
588, 57bnj1023 28489 1  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f `  i
)  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    \ cdif 3260    C_ wss 3263   (/)c0 3571   {csn 3757   U_ciun 4035   class class class wbr 4153    _E cep 4433   suc csuc 4524   omcom 4785   dom cdm 4818    Fn wfn 5389   ` cfv 5394    /\ w-bnj17 28388    predc-bnj14 28390    TrFow-bnj19 28398
This theorem is referenced by:  bnj1030  28694
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-iota 5358  df-fn 5397  df-fv 5402  df-bnj17 28389  df-bnj19 28399
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