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Theorem bnj594 29345
Description: Technical lemma for bnj852 29354. 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 974 . . . . . . . 8  |-  ( ch 
->  ph )
3 bnj594.1 . . . . . . . 8  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
42, 3sylib 190 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
5 bnj594.11 . . . . . . . . 9  |-  ( ch'  <->  (
g  Fn  n  /\  ph' 
/\  ps' ) )
65simp2bi 974 . . . . . . . 8  |-  ( ch'  ->  ph' )
7 bnj594.9 . . . . . . . 8  |-  ( ph'  <->  (
g `  (/) )  = 
pred ( x ,  A ,  R ) )
86, 7sylib 190 . . . . . . 7  |-  ( ch'  ->  ( g `  (/) )  = 
pred ( x ,  A ,  R ) )
9 eqtr3 2457 . . . . . . 7  |-  ( ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ( g `  (/) )  =  pred (
x ,  A ,  R ) )  -> 
( f `  (/) )  =  ( g `  (/) ) )
104, 8, 9syl2an 465 . . . . . 6  |-  ( ( ch  /\  ch' )  -> 
( f `  (/) )  =  ( g `  (/) ) )
11103adant1 976 . . . . 5  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  (/) )  =  ( g `  (/) ) )
12 fveq2 5730 . . . . . 6  |-  ( j  =  (/)  ->  ( f `
 j )  =  ( f `  (/) ) )
13 fveq2 5730 . . . . . 6  |-  ( j  =  (/)  ->  ( g `
 j )  =  ( g `  (/) ) )
1412, 13eqeq12d 2452 . . . . 5  |-  ( j  =  (/)  ->  ( ( f `  j )  =  ( g `  j )  <->  ( f `  (/) )  =  ( g `  (/) ) ) )
1511, 14syl5ibr 214 . . . 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 205 . . 3  |-  ( j  =  (/)  ->  th )
1817a1d 24 . 2  |-  ( j  =  (/)  ->  ( ( j  e.  n  /\  ta )  ->  th )
)
19 bnj253 29130 . . . . . 6  |-  ( ( n  e.  D  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( (
n  e.  D  /\  n  e.  D )  /\  ch  /\  ch' ) )
20 bnj252 29129 . . . . . 6  |-  ( ( n  e.  D  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( n  e.  D  /\  (
n  e.  D  /\  ch  /\  ch' ) ) )
21 anidm 627 . . . . . . 7  |-  ( ( n  e.  D  /\  n  e.  D )  <->  n  e.  D )
22213anbi1i 1145 . . . . . 6  |-  ( ( ( n  e.  D  /\  n  e.  D
)  /\  ch  /\  ch' )  <->  ( n  e.  D  /\  ch  /\  ch' ) )
2319, 20, 223bitr3i 268 . . . . 5  |-  ( ( n  e.  D  /\  ( n  e.  D  /\  ch  /\  ch' ) )  <-> 
( n  e.  D  /\  ch  /\  ch' ) )
24 df-bnj17 29113 . . . . . . . . . 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 29214 . . . . . . . . . . 11  |-  ( ta 
->  A. k ta )
2726bnj1352 29261 . . . . . . . . . 10  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  /\  ta )  ->  A. k ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  /\  ta ) )
2824, 27hbxfrbi 1578 . . . . . . . . 9  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  A. k ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )
)
29 bnj170 29124 . . . . . . . . . . . 12  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  <->  ( (
j  e.  n  /\  n  e.  D )  /\  j  =/=  (/) ) )
30 bnj594.7 . . . . . . . . . . . . . . 15  |-  D  =  ( om  \  { (/)
} )
3130bnj923 29199 . . . . . . . . . . . . . 14  |-  ( n  e.  D  ->  n  e.  om )
32 elnn 4857 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
3331, 32sylan2 462 . . . . . . . . . . . . 13  |-  ( ( j  e.  n  /\  n  e.  D )  ->  j  e.  om )
3433anim1i 553 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D
)  /\  j  =/=  (/) )  ->  ( j  e.  om  /\  j  =/=  (/) ) )
3529, 34sylbi 189 . . . . . . . . . . 11  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  ->  (
j  e.  om  /\  j  =/=  (/) ) )
36 nnsuc 4864 . . . . . . . . . . 11  |-  ( ( j  e.  om  /\  j  =/=  (/) )  ->  E. k  e.  om  j  =  suc  k )
37 rexex 2767 . . . . . . . . . . 11  |-  ( E. k  e.  om  j  =  suc  k  ->  E. k 
j  =  suc  k
)
3835, 36, 373syl 19 . . . . . . . . . 10  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D )  ->  E. k 
j  =  suc  k
)
3938bnj721 29187 . . . . . . . . 9  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k  j  =  suc  k )
4028, 39bnj596 29176 . . . . . . . 8  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
41 bnj667 29182 . . . . . . . . . . 11  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  ( j  e.  n  /\  n  e.  D  /\  ta ) )
4241anim1i 553 . . . . . . . . . 10  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k )  ->  (
( j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
43 bnj258 29134 . . . . . . . . . 10  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  <->  ( ( j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k ) )
4442, 43sylibr 205 . . . . . . . . 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 29113 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  <->  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta ) )
46 bnj219 29162 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  suc  k  -> 
k  _E  j )
47463ad2ant3 981 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  k  _E  j
)
4847adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  k  _E  j
)
49 vex 2961 . . . . . . . . . . . . . . . . . . 19  |-  k  e. 
_V
5049bnj216 29161 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  suc  k  -> 
k  e.  j )
51 df-3an 939 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  e.  j  /\  j  e.  n  /\  n  e.  D )  <->  ( ( k  e.  j  /\  j  e.  n
)  /\  n  e.  D ) )
52 3anrot 942 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  e.  j  /\  j  e.  n  /\  n  e.  D )  <->  ( j  e.  n  /\  n  e.  D  /\  k  e.  j )
)
53 ancom 439 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( k  e.  j  /\  j  e.  n
)  /\  n  e.  D )  <->  ( n  e.  D  /\  (
k  e.  j  /\  j  e.  n )
) )
5451, 52, 533bitr3i 268 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  n  e.  D  /\  k  e.  j )  <->  ( n  e.  D  /\  ( k  e.  j  /\  j  e.  n
) ) )
55 eldifi 3471 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  ( om  \  { (/)
} )  ->  n  e.  om )
5655, 30eleq2s 2530 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
57 nnord 4855 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  om  ->  Ord  n )
58 ordtr1 4626 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Ord  n  ->  ( (
k  e.  j  /\  j  e.  n )  ->  k  e.  n ) )
5956, 57, 583syl 19 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  D  ->  (
( k  e.  j  /\  j  e.  n
)  ->  k  e.  n ) )
6059imp 420 . . . . . . . . . . . . . . . . . . 19  |-  ( ( n  e.  D  /\  ( k  e.  j  /\  j  e.  n
) )  ->  k  e.  n )
6154, 60sylbi 189 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  n  /\  n  e.  D  /\  k  e.  j )  ->  k  e.  n )
6250, 61syl3an3 1220 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  k  e.  n
)
63 rsp 2768 . . . . . . . . . . . . . . . . . 18  |-  ( A. k  e.  n  (
k  _E  j  ->  [. k  /  j ]. th )  ->  (
k  e.  n  -> 
( k  _E  j  ->  [. k  /  j ]. th ) ) )
6425, 63sylbi 189 . . . . . . . . . . . . . . . . 17  |-  ( ta 
->  ( k  e.  n  ->  ( k  _E  j  ->  [. k  /  j ]. th ) ) )
6562, 64mpan9 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  ( k  _E  j  ->  [. k  / 
j ]. th ) )
6648, 65mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ta )  ->  [. k  /  j ]. th )
6745, 66sylbi 189 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  [. k  /  j ]. th )
6867anim1i 553 . . . . . . . . . . . . 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 29129 . . . . . . . . . . . . 13  |-  ( (
[. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' )  <->  ( [. k  /  j ]. th  /\  ( n  e.  D  /\  ch  /\  ch' ) ) )
7068, 69sylibr 205 . . . . . . . . . . . 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 29143 . . . . . . . . . . . . 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 pm3.35 572 . . . . . . . . . . . . . 14  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  /\  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )  ->  (
f `  k )  =  ( g `  k ) )
7472, 73sylan2b 463 . . . . . . . . . . . . 13  |-  ( ( ( n  e.  D  /\  ch  /\  ch' )  /\  [. k  /  j ]. th )  ->  ( f `
 k )  =  ( g `  k
) )
7571, 74sylbi 189 . . . . . . . . . . . 12  |-  ( (
[. k  /  j ]. th  /\  n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) )
76 iuneq1 4108 . . . . . . . . . . . 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 19 . . . . . . . . . . 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 29181 . . . . . . . . . . . . 13  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k ) )
791simp3bi 975 . . . . . . . . . . . . . 14  |-  ( ch 
->  ps )
805simp3bi 975 . . . . . . . . . . . . . 14  |-  ( ch'  ->  ps' )
8179, 80bnj240 29125 . . . . . . . . . . . . 13  |-  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( ps  /\  ps' ) )
8278, 81anim12i 551 . . . . . . . . . . . 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 445 . . . . . . . . . . . . 13  |-  ( ( ps  /\  ps' )  ->  ps )
8483anim2i 554 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ( ps  /\  ps' ) )  -> 
( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps ) )
85 simp3 960 . . . . . . . . . . . . . 14  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  ->  j  =  suc  k )
8685anim1i 553 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps )  ->  ( j  =  suc  k  /\  ps ) )
87 simpl1 961 . . . . . . . . . . . . . 14  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  (
j  =  suc  k  /\  ps ) )  -> 
j  e.  n )
88 df-3an 939 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  <-> 
( ( j  e.  n  /\  n  e.  D )  /\  j  =  suc  k ) )
89 ancom 439 . . . . . . . . . . . . . . . . 17  |-  ( ( ( j  e.  n  /\  n  e.  D
)  /\  j  =  suc  k )  <->  ( j  =  suc  k  /\  (
j  e.  n  /\  n  e.  D )
) )
9088, 89bitri 242 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  <-> 
( j  =  suc  k  /\  ( j  e.  n  /\  n  e.  D ) ) )
91 elnn 4857 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  j  /\  j  e.  om )  ->  k  e.  om )
9250, 33, 91syl2an 465 . . . . . . . . . . . . . . . 16  |-  ( ( j  =  suc  k  /\  ( j  e.  n  /\  n  e.  D
) )  ->  k  e.  om )
9390, 92sylbi 189 . . . . . . . . . . . . . . 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 29342 . . . . . . . . . . . . . . . 16  |-  ( ps  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( f `  suc  k )  =  U_ y  e.  ( f `  k )  pred (
y ,  A ,  R ) ) )
9695bnj590 29343 . . . . . . . . . . . . . . 15  |-  ( ( j  =  suc  k  /\  ps )  ->  (
k  e.  om  ->  ( j  e.  n  -> 
( f `  j
)  =  U_ y  e.  ( f `  k
)  pred ( y ,  A ,  R ) ) ) )
9793, 96mpan9 457 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 458 . . . . . . . . . . . 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 19 . . . . . . . . . . 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 449 . . . . . . . . . . . . 13  |-  ( ( ps  /\  ps' )  ->  ps' )
102101anim2i 554 . . . . . . . . . . . 12  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ( ps  /\  ps' ) )  -> 
( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps' ) )
10385anim1i 553 . . . . . . . . . . . . 13  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k )  /\  ps' )  -> 
( j  =  suc  k  /\  ps' ) )
104 simpl1 961 . . . . . . . . . . . . . 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 29342 . . . . . . . . . . . . . . . 16  |-  ( ps'  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( g `  suc  k )  =  U_ y  e.  ( g `  k )  pred (
y ,  A ,  R ) ) )
107106bnj590 29343 . . . . . . . . . . . . . . 15  |-  ( ( j  =  suc  k  /\  ps' )  ->  ( k  e.  om  ->  (
j  e.  n  -> 
( g `  j
)  =  U_ y  e.  ( g `  k
)  pred ( y ,  A ,  R ) ) ) )
10893, 107mpan9 457 . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . 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 458 . . . . . . . . . . . 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 19 . . . . . . . . . . 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 2480 . . . . . . . . . 10  |-  ( ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  ( f `  j )  =  ( g `  j ) )
113112ex 425 . . . . . . . . 9  |-  ( ( j  e.  n  /\  n  e.  D  /\  j  =  suc  k  /\  ta )  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
11444, 113syl 16 . . . . . . . 8  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  /\  j  =  suc  k )  ->  (
( n  e.  D  /\  ch  /\  ch' )  -> 
( f `  j
)  =  ( g `
 j ) ) )
11540, 114bnj593 29175 . . . . . . 7  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  ->  E. k ( ( n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
116 bnj258 29134 . . . . . . 7  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  n  e.  D  /\  ta )  <->  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  /\  n  e.  D
) )
117 19.9v 1677 . . . . . . 7  |-  ( E. k ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `
 j )  =  ( g `  j
) )  <->  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
118115, 116, 1173imtr3i 258 . . . . . 6  |-  ( ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  /\  n  e.  D
)  ->  ( (
n  e.  D  /\  ch  /\  ch' )  ->  (
f `  j )  =  ( g `  j ) ) )
119118expimpd 588 . . . . 5  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  ( ( n  e.  D  /\  ( n  e.  D  /\  ch  /\  ch' ) )  ->  (
f `  j )  =  ( g `  j ) ) )
12023, 119syl5bir 211 . . . 4  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )
121120, 16sylibr 205 . . 3  |-  ( ( j  =/=  (/)  /\  j  e.  n  /\  ta )  ->  th )
1221213expib 1157 . 2  |-  ( j  =/=  (/)  ->  ( (
j  e.  n  /\  ta )  ->  th )
)
12318, 122pm2.61ine 2682 1  |-  ( ( j  e.  n  /\  ta )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   [.wsbc 3163    \ cdif 3319   (/)c0 3630   {csn 3816   U_ciun 4095   class class class wbr 4214    _E cep 4494   Ord word 4582   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29112    predc-bnj14 29114
This theorem is referenced by:  bnj580  29346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-iota 5420  df-fv 5464  df-bnj17 29113
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