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Theorem bnj1145 28785
Description: Technical lemma for bnj69 28802. 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
bnj1145.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1145.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1145.3  |-  D  =  ( om  \  { (/)
} )
bnj1145.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1145.5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1145.6  |-  ( th  <->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
Assertion
Ref Expression
bnj1145  |-  trCl ( X ,  A ,  R )  C_  A
Distinct variable groups:    A, f,
i, j, n, y    D, i, j    R, f, i, j, n, y   
f, X, i, n, y    ch, j    ph, i
Allowed substitution hints:    ph( y, f, j, n)    ps( y,
f, i, j, n)    ch( y, f, i, n)    th( y, f, i, j, n)    B( y, f, i, j, n)    D( y,
f, n)    X( j)

Proof of Theorem bnj1145
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1145.1 . . 3  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1145.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1145.3 . . 3  |-  D  =  ( om  \  { (/)
} )
4 bnj1145.4 . . 3  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
51, 2, 3, 4bnj882 28720 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
6 ss2iun 4001 . . . 4  |-  ( A. f  e.  B  U_ i  e.  dom  f ( f `
 i )  C_  A  ->  U_ f  e.  B  U_ i  e.  dom  f
( f `  i
)  C_  U_ f  e.  B  A )
7 bnj1145.5 . . . . . . 7  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
87, 4bnj1083 28770 . . . . . 6  |-  ( f  e.  B  <->  E. n ch )
92bnj1095 28575 . . . . . . . . . 10  |-  ( ps 
->  A. i ps )
109, 7bnj1096 28576 . . . . . . . . 9  |-  ( ch 
->  A. i ch )
1110nfi 1551 . . . . . . . 8  |-  F/ i ch
123bnj1098 28577 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
137bnj1232 28598 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  n  e.  D
)
14133anim3i 1139 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
1512, 14bnj1101 28578 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
j  e.  n  /\  i  =  suc  j ) )
16 ancl 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) ) )
1715, 16bnj101 28511 . . . . . . . . . . . . . . 15  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
18 bnj1145.6 . . . . . . . . . . . . . . . . 17  |-  ( th  <->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
1918imbi2i 303 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )  <->  ( (
i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
2019exbii 1582 . . . . . . . . . . . . . . 15  |-  ( E. j ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )  <->  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) ) )
2117, 20mpbir 200 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )
22 bnj213 28676 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
2322bnj226 28524 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
24 simpr 447 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  i  =  suc  j )
2518, 24bnj833 28550 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  =  suc  j )
26 simp2 956 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  n )
27133ad2ant3 978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  n  e.  D )
283bnj923 28560 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
29 elnn 4748 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
3028, 29sylan2 460 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  n  /\  n  e.  D )  ->  i  e.  om )
3126, 27, 30syl2anc 642 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  om )
3218, 31bnj832 28549 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  om )
33 vex 2867 . . . . . . . . . . . . . . . . . . . 20  |-  j  e. 
_V
3433bnj216 28522 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  suc  j  -> 
j  e.  i )
35 elnn 4748 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  i  /\  i  e.  om )  ->  j  e.  om )
3634, 35sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =  suc  j  /\  i  e.  om )  ->  j  e.  om )
3725, 32, 36syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  j  e.  om )
3818, 26bnj832 28549 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  n
)
3925, 38eqeltrrd 2433 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  suc  j  e.  n
)
402bnj589 28703 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4140biimpi 186 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps 
->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4241bnj708 28547 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `
 suc  j )  =  U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
43 rsp 2679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  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 ) ) ) )
4442, 43syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  ->  (
j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) ) )
457, 44sylbi 187 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
46453ad2ant3 978 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
4718, 46bnj832 28549 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
4837, 39, 47mp2d 41 . . . . . . . . . . . . . . . 16  |-  ( th 
->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
49 fveq2 5608 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
5049eqeq1d 2366 . . . . . . . . . . . . . . . . 17  |-  ( 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 ) ) )
5125, 50syl 15 . . . . . . . . . . . . . . . 16  |-  ( th 
->  ( ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R )  <->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
5248, 51mpbird 223 . . . . . . . . . . . . . . 15  |-  ( th 
->  ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
5323, 52bnj1262 28605 . . . . . . . . . . . . . 14  |-  ( th 
->  ( f `  i
)  C_  A )
5421, 53bnj1023 28574 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
f `  i )  C_  A )
55 3anass 938 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  <->  ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) ) )
5655imbi1i 315 . . . . . . . . . . . . . 14  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5756exbii 1582 . . . . . . . . . . . . 13  |-  ( E. j ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  E. j ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5854, 57mpbi 199 . . . . . . . . . . . 12  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch )
)  ->  ( f `  i )  C_  A
)
591biimpi 186 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
607, 59bnj771 28556 . . . . . . . . . . . . . 14  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
61 fveq2 5608 . . . . . . . . . . . . . . 15  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
62 bnj213 28676 . . . . . . . . . . . . . . . 16  |-  pred ( X ,  A ,  R )  C_  A
63 sseq1 3275 . . . . . . . . . . . . . . . 16  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( ( f `  (/) )  C_  A  <->  pred ( X ,  A ,  R
)  C_  A )
)
6462, 63mpbiri 224 . . . . . . . . . . . . . . 15  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  C_  A )
65 sseq1 3275 . . . . . . . . . . . . . . . 16  |-  ( ( f `  i )  =  ( f `  (/) )  ->  ( (
f `  i )  C_  A  <->  ( f `  (/) )  C_  A )
)
6665biimpar 471 . . . . . . . . . . . . . . 15  |-  ( ( ( f `  i
)  =  ( f `
 (/) )  /\  (
f `  (/) )  C_  A )  ->  (
f `  i )  C_  A )
6761, 64, 66syl2an 463 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  C_  A
)
6860, 67sylan2 460 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  ch )  ->  ( f `  i )  C_  A
)
6968adantrl 696 . . . . . . . . . . . 12  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A )
7058, 69bnj1109 28580 . . . . . . . . . . 11  |-  E. j
( ( i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )
71 19.9v 1664 . . . . . . . . . . 11  |-  ( E. j ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
)  <->  ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
) )
7270, 71mpbi 199 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ch )  ->  ( f `
 i )  C_  A )
7372expcom 424 . . . . . . . . 9  |-  ( ch 
->  ( i  e.  n  ->  ( f `  i
)  C_  A )
)
74 fndm 5425 . . . . . . . . . . 11  |-  ( f  Fn  n  ->  dom  f  =  n )
757, 74bnj770 28555 . . . . . . . . . 10  |-  ( ch 
->  dom  f  =  n )
76 eleq2 2419 . . . . . . . . . . 11  |-  ( dom  f  =  n  -> 
( i  e.  dom  f 
<->  i  e.  n ) )
7776imbi1d 308 . . . . . . . . . 10  |-  ( dom  f  =  n  -> 
( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7875, 77syl 15 . . . . . . . . 9  |-  ( ch 
->  ( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7973, 78mpbird 223 . . . . . . . 8  |-  ( ch 
->  ( i  e.  dom  f  ->  ( f `  i )  C_  A
) )
8011, 79ralrimi 2700 . . . . . . 7  |-  ( ch 
->  A. i  e.  dom  f ( f `  i )  C_  A
)
8180exlimiv 1634 . . . . . 6  |-  ( E. n ch  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
828, 81sylbi 187 . . . . 5  |-  ( f  e.  B  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
83 ss2iun 4001 . . . . . 6  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  U_ i  e.  dom  f  A )
84 bnj1143 28584 . . . . . 6  |-  U_ i  e.  dom  f  A  C_  A
8583, 84syl6ss 3267 . . . . 5  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
8682, 85syl 15 . . . 4  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
876, 86mprg 2688 . . 3  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  U_ f  e.  B  A
884bnj1317 28616 . . . 4  |-  ( w  e.  B  ->  A. f  w  e.  B )
8988bnj1146 28585 . . 3  |-  U_ f  e.  B  A  C_  A
9087, 89sstri 3264 . 2  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  A
915, 90eqsstri 3284 1  |-  trCl ( X ,  A ,  R )  C_  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   A.wral 2619   E.wrex 2620    \ cdif 3225    C_ wss 3228   (/)c0 3531   {csn 3716   U_ciun 3986   suc csuc 4476   omcom 4738   dom cdm 4771    Fn wfn 5332   ` cfv 5337    /\ w-bnj17 28473    predc-bnj14 28475    trClc-bnj18 28481
This theorem is referenced by:  bnj1147  28786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-iota 5301  df-fn 5340  df-fv 5345  df-bnj17 28474  df-bnj14 28476  df-bnj18 28482
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