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Theorem bnj594 29260
Description: Technical lemma for bnj852 29269. 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
bnj594.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj594.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj594.3  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
bnj594.7  |-  D  =  ( om  \  { (/)
} )
bnj594.9  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj594.10  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) )
bnj594.11  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
bnj594.15  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
bnj594.16  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
bnj594.17  |-  ( ta  <->  A. k  e.  n  ( k  _E  j  ->  [. k  /  j ]. th ) )
Assertion
Ref Expression
bnj594  |-  ( ( j  e.  n  /\  ta )  ->  th )
Distinct variable groups:    A, i,
k    D, k    R, i, k    ch, k    k, ch'    f, i, k, y    g,
i, k, y    i, n, k    j, k
Allowed substitution hints:    ph( x, y, f, g, i, j, k, n)    ps( x, y, f, g, i, j, k, n)    ch( x, y, f, g, i, j, n)    th( x, y, f, g, i, j, k, n)    ta( x, y, f, g, i, j, k, n)    A( x, y, f, g, j, n)    D( x, y, f, g, i, j, n)    R( x, y, f, g, j, n)    ph'( x, y, f, g, i, j, k, n)    ps'( x, y, f, g, i, j, k, n)    ch'( x, y, f, g, i, j, n)

Proof of Theorem bnj594
StepHypRef Expression
1 bnj594.3 . . . . . . . . 9  |-  ( ch  <->  ( f  Fn  n  /\  ph 
/\  ps ) )
21simp2bi 971 . . . . . . . 8  |-  ( ch 
->  ph )
3 bnj594.1 . . . . . . . 8  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
42, 3sylib 188 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
5 bnj594.11 . . . . . . . . 9  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
65simp2bi 971 . . . . . . . 8  |-  ( ch'  ->  ph' )
7 bnj594.9 . . . . . . . 8  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
86, 7sylib 188 . . . . . . 7  |-  ( ch'  ->  ( g `  (/) )  = 
pred ( x ,  A ,  R ) )
9 eqtr3 2315 . . . . . . 7  |-  ( ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ( g `  (/) )  =  pred (
x ,  A ,  R ) )  -> 
( f `  (/) )  =  ( g `  (/) ) )
104, 8, 9syl2an 463 . . . . . 6  |-  ( ( ch  /\  ch' )  -> 
( f `  (/) )  =  ( g `  (/) ) )
11103adant1 973 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  (/) )  =  ( g `  (/) ) )
12 fveq2 5541 . . . . . 6  |-  ( j  =  (/)  ->  ( f `
 j )  =  ( f `  (/) ) )
13 fveq2 5541 . . . . . 6  |-  ( j  =  (/)  ->  ( g `
 j )  =  ( g `  (/) ) )
1412, 13eqeq12d 2310 . . . . 5  |-  ( j  =  (/)  ->  ( ( f `  j )  =  ( g `  j )  <->  ( f `  (/) )  =  ( g `  (/) ) ) )
1511, 14syl5ibr 212 . . . 4  |-  ( j  =  (/)  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
16 bnj594.15 . . . 4  |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
1715, 16sylibr 203 . . 3  |-  ( j  =  (/)  ->  th )
1817a1d 22 . 2  |-  ( j  =  (/)  ->  ( ( j  e.  n  /\  ta )  ->  th )
)
19 bnj253 29045 . . . . . 6  |-  ( ( n  e.  D  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( (
n  e.  D  /\  n  e.  D )  /\  ch  /\  ch' ) )
20 bnj252 29044 . . . . . 6  |-  ( ( n  e.  D  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( n  e.  D  /\  (
n  e.  D  /\  ch  /\  ch' ) ) )
21 anidm 625 . . . . . . 7  |-  ( ( n  e.  D  /\  n  e.  D )  <->  n  e.  D )
22213anbi1i 1142 . . . . . 6  |-  ( ( ( n  e.  D  /\  n  e.  D
)  /\  ch  /\  ch' )  <->  ( n  e.  D  /\  ch  /\  ch' ) )
2319, 20, 223bitr3i 266 . . . . 5  |-  ( ( n  e.  D  /\  ( n  e.  D  /\  ch  /\  ch' ) )  <-> 
( n  e.  D  /\  ch  /\  ch' ) )
24 df-bnj17 29028 . . . . . . . . . 10  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  <->  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  /\  ta ) )
25 bnj594.17 . . . . . . . . . . . 12  |-  ( ta  <->  A. k  e.  n  ( k  _E  j  ->  [. k  /  j ]. th ) )
2625bnj1095 29129 . . . . . . . . . . 11  |-  ( ta 
->  A. k ta )
2726bnj1352 29176 . . . . . . . . . 10  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  /\  ta )  ->  A. k ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  /\  ta ) )
2824, 27hbxfrbi 1558 . . . . . . . . 9  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  A. k ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )
)
29 bnj170 29039 . . . . . . . . . . . 12  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  <->  ( (
j  e.  n  /\  n  e.  D )  /\  j  =/=  (/) ) )
30 bnj594.7 . . . . . . . . . . . . . . 15  |-  D  =  ( om  \  { (/)
} )
3130bnj923 29114 . . . . . . . . . . . . . 14  |-  ( n  e.  D  ->  n  e.  om )
32 elnn 4682 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
3331, 32sylan2 460 . . . . . . . . . . . . 13  |-  ( ( j  e.  n  /\  n  e.  D )  ->  j  e.  om )
3433anim1i 551 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D
)  /\  j  =/=  (/) )  ->  ( j  e.  om  /\  j  =/=  (/) ) )
3529, 34sylbi 187 . . . . . . . . . . 11  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  ->  (
j  e.  om  /\  j  =/=  (/) ) )
36 nnsuc 4689 . . . . . . . . . . 11  |-  ( ( j  e.  om  /\  j  =/=  (/) )  ->  E. k  e.  om  j  =  suc  k )
37 rexex 2615 . . . . . . . . . . 11  |-  ( E. k  e.  om  j  =  suc  k  ->  E. k 
j  =  suc  k
)
3835, 36, 373syl 18 . . . . . . . . . 10  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  ->  E. k 
j  =  suc  k
)
3938bnj721 29102 . . . . . . . . 9  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k  j  =  suc  k )
4028, 39bnj596 29091 . . . . . . . 8  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
41 bnj667 29097 . . . . . . . . . . 11  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  ( j  e.  n  /\  n  e.  D  /\  ta ) )
4241anim1i 551 . . . . . . . . . 10  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k )  ->  (
( j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
43 bnj258 29049 . . . . . . . . . 10  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  <->  ( ( j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
4442, 43sylibr 203 . . . . . . . . 9  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k )  ->  (
j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta ) )
45 df-bnj17 29028 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  <->  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta ) )
46 bnj219 29077 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  suc  k  -> 
k  _E  j )
47463ad2ant3 978 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  k  _E  j
)
4847adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  k  _E  j
)
49 vex 2804 . . . . . . . . . . . . . . . . . . 19  |-  k  e. 
_V
5049bnj216 29076 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  suc  k  -> 
k  e.  j )
51 df-3an 936 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  e.  j  /\  j  e.  n  /\  n  e.  D )  <->  ( ( k  e.  j  /\  j  e.  n
)  /\  n  e.  D ) )
52 3anrot 939 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  e.  j  /\  j  e.  n  /\  n  e.  D )  <->  ( j  e.  n  /\  n  e.  D  /\  k  e.  j )
)
53 ancom 437 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( k  e.  j  /\  j  e.  n
)  /\  n  e.  D )  <->  ( n  e.  D  /\  (
k  e.  j  /\  j  e.  n )
) )
5451, 52, 533bitr3i 266 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  n  e.  D  /\  k  e.  j )  <->  ( n  e.  D  /\  ( k  e.  j  /\  j  e.  n
) ) )
55 eldifi 3311 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  ( om  \  { (/)
} )  ->  n  e.  om )
5655, 30eleq2s 2388 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
57 nnord 4680 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  om  ->  Ord  n )
58 ordtr1 4451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Ord  n  ->  ( (
k  e.  j  /\  j  e.  n )  ->  k  e.  n ) )
5956, 57, 583syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  D  ->  (
( k  e.  j  /\  j  e.  n
)  ->  k  e.  n ) )
6059imp 418 . . . . . . . . . . . . . . . . . . 19  |-  ( ( n  e.  D  /\  ( k  e.  j  /\  j  e.  n
) )  ->  k  e.  n )
6154, 60sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  n  /\  n  e.  D  /\  k  e.  j )  ->  k  e.  n )
6250, 61syl3an3 1217 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  k  e.  n
)
63 rsp 2616 . . . . . . . . . . . . . . . . . 18  |-  ( A. k  e.  n  (
k  _E  j  ->  [. k  /  j ]. th )  ->  (
k  e.  n  -> 
( k  _E  j  ->  [. k  /  j ]. th ) ) )
6425, 63sylbi 187 . . . . . . . . . . . . . . . . 17  |-  ( ta 
->  ( k  e.  n  ->  ( k  _E  j  ->  [. k  /  j ]. th ) ) )
6562, 64mpan9 455 . . . . . . . . . . . . . . . 16  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  ( k  _E  j  ->  [. k  / 
j ]. th ) )
6648, 65mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  [. k  /  j ]. th )
6745, 66sylbi 187 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  [. k  /  j ]. th )
6867anim1i 551 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( [. k  /  j ]. th  /\  ( n  e.  D  /\  ch  /\  ch' ) ) )
69 bnj252 29044 . . . . . . . . . . . . 13  |-  ( (
[. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( [. k  /  j ]. th  /\  ( n  e.  D  /\  ch  /\  ch' ) ) )
7068, 69sylibr 203 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( [. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' ) )
71 bnj446 29058 . . . . . . . . . . . . 13  |-  ( (
[. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( ( n  e.  D  /\  ch  /\  ch' )  /\  [. k  /  j ]. th ) )
72 bnj594.16 . . . . . . . . . . . . . 14  |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  k
)  =  ( g `
 k ) ) )
73 pm3.35 570 . . . . . . . . . . . . . 14  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  /\  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )  ->  (
f `  k )  =  ( g `  k ) )
7472, 73sylan2b 461 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  /\  [. k  /  j ]. th )  ->  ( f `
 k )  =  ( g `  k
) )
7571, 74sylbi 187 . . . . . . . . . . . 12  |-  ( (
[. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) )
76 iuneq1 3934 . . . . . . . . . . . 12  |-  ( ( f `  k )  =  ( g `  k )  ->  U_ y  e.  ( f `  k
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) )
7770, 75, 763syl 18 . . . . . . . . . . 11  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  U_ y  e.  ( f `  k ) 
pred ( y ,  A ,  R )  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) )
78 bnj658 29096 . . . . . . . . . . . . 13  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k ) )
791simp3bi 972 . . . . . . . . . . . . . 14  |-  ( ch 
->  ps )
805simp3bi 972 . . . . . . . . . . . . . 14  |-  ( ch'  ->  ps' )
8179, 80bnj240 29040 . . . . . . . . . . . . 13  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( ps  /\  ps' ) )
8278, 81anim12i 549 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ( ps  /\  ps' ) ) )
83 simpl 443 . . . . . . . . . . . . 13  |-  ( ( ps  /\  ps' )  ->  ps )
8483anim2i 552 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ( ps  /\  ps' ) )  -> 
( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps ) )
85 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  j  =  suc  k )
8685anim1i 551 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps )  ->  ( j  =  suc  k  /\  ps ) )
87 simpl1 958 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps ) )  -> 
j  e.  n )
88 df-3an 936 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  <-> 
( ( j  e.  n  /\  n  e.  D )  /\  j  =  suc  k ) )
89 ancom 437 . . . . . . . . . . . . . . . . 17  |-  ( ( ( j  e.  n  /\  n  e.  D
)  /\  j  =  suc  k )  <->  ( j  =  suc  k  /\  (
j  e.  n  /\  n  e.  D )
) )
9088, 89bitri 240 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  <-> 
( j  =  suc  k  /\  ( j  e.  n  /\  n  e.  D ) ) )
91 elnn 4682 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  j  /\  j  e.  om )  ->  k  e.  om )
9250, 33, 91syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( j  =  suc  k  /\  ( j  e.  n  /\  n  e.  D
) )  ->  k  e.  om )
9390, 92sylbi 187 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  k  e.  om )
94 bnj594.2 . . . . . . . . . . . . . . . . 17  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
9594bnj589 29257 . . . . . . . . . . . . . . . 16  |-  ( ps  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( f `  suc  k )  =  U_ y  e.  ( f `  k )  pred (
y ,  A ,  R ) ) )
9695bnj590 29258 . . . . . . . . . . . . . . 15  |-  ( ( j  =  suc  k  /\  ps )  ->  (
k  e.  om  ->  ( j  e.  n  -> 
( f `  j
)  =  U_ y  e.  ( f `  k
)  pred ( y ,  A ,  R ) ) ) )
9793, 96mpan9 455 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps ) )  -> 
( j  e.  n  ->  ( f `  j
)  =  U_ y  e.  ( f `  k
)  pred ( y ,  A ,  R ) ) )
9887, 97mpd 14 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps ) )  -> 
( f `  j
)  =  U_ y  e.  ( f `  k
)  pred ( y ,  A ,  R ) )
9986, 98syldan 456 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps )  ->  ( f `  j )  =  U_ y  e.  ( f `  k )  pred (
y ,  A ,  R ) )
10082, 84, 993syl 18 . . . . . . . . . . 11  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( f `  j )  =  U_ y  e.  ( f `  k )  pred (
y ,  A ,  R ) )
101 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ps  /\  ps' )  ->  ps' )
102101anim2i 552 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ( ps  /\  ps' ) )  -> 
( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps' ) )
10385anim1i 551 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps' )  -> 
( j  =  suc  k  /\  ps' ) )
104 simpl1 958 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps' ) )  ->  j  e.  n )
105 bnj594.10 . . . . . . . . . . . . . . . . 17  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) )
106105bnj589 29257 . . . . . . . . . . . . . . . 16  |-  ( ps'  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( g `  suc  k )  =  U_ y  e.  ( g `  k )  pred (
y ,  A ,  R ) ) )
107106bnj590 29258 . . . . . . . . . . . . . . 15  |-  ( ( j  =  suc  k  /\  ps' )  ->  ( k  e.  om  ->  (
j  e.  n  -> 
( g `  j
)  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) ) ) )
10893, 107mpan9 455 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps' ) )  ->  (
j  e.  n  -> 
( g `  j
)  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) ) )
109104, 108mpd 14 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps' ) )  ->  (
g `  j )  =  U_ y  e.  ( g `  k ) 
pred ( y ,  A ,  R ) )
110103, 109syldan 456 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps' )  -> 
( g `  j
)  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) )
11182, 102, 1103syl 18 . . . . . . . . . . 11  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( g `  j )  =  U_ y  e.  ( g `  k )  pred (
y ,  A ,  R ) )
11277, 100, 1113eqtr4d 2338 . . . . . . . . . 10  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( f `  j )  =  ( g `  j ) )
113112ex 423 . . . . . . . . 9  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
11444, 113syl 15 . . . . . . . 8  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k )  ->  (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
11540, 114bnj593 29090 . . . . . . 7  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
116 bnj258 29049 . . . . . . 7  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  <->  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  /\  n  e.  D
) )
117 19.9v 1653 . . . . . . 7  |-  ( E. k ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `
 j )  =  ( g `  j
) )  <->  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
118115, 116, 1173imtr3i 256 . . . . . 6  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  /\  n  e.  D
)  ->  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
119118expimpd 586 . . . . 5  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  ( ( n  e.  D  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  (
f `  j )  =  ( g `  j ) ) )
12023, 119syl5bir 209 . . . 4  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )
121120, 16sylibr 203 . . 3  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  th )
1221213expib 1154 . 2  |-  ( j  =/=  (/)  ->  ( (
j  e.  n  /\  ta )  ->  th )
)
12318, 122pm2.61ine 2535 1  |-  ( ( j  e.  n  /\  ta )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   [.wsbc 3004    \ cdif 3162   (/)c0 3468   {csn 3653   U_ciun 3921   class class class wbr 4039    _E cep 4319   Ord word 4407   suc csuc 4410   omcom 4672    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029
This theorem is referenced by:  bnj580  29261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-iota 5235  df-fv 5279  df-bnj17 29028
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