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Theorem bnj1145 31818
Description: Technical lemma for bnj69 31835. 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 31753 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
6 ss2iun 4183 . . . 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 31803 . . . . . 6  |-  ( f  e.  B  <->  E. n ch )
92bnj1095 31609 . . . . . . . . . 10  |-  ( ps 
->  A. i ps )
109, 7bnj1096 31610 . . . . . . . . 9  |-  ( ch 
->  A. i ch )
1110nfi 1601 . . . . . . . 8  |-  F/ i ch
123bnj1098 31611 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
137bnj1232 31631 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  n  e.  D
)
14133anim3i 1170 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
1512, 14bnj1101 31612 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
j  e.  n  /\  i  =  suc  j ) )
16 ancl 543 . . . . . . . . . . . . . . . 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 31546 . . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )  <->  ( (
i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
2019exbii 1639 . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )
22 bnj213 31709 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
2322bnj226 31559 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
24 simpr 458 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  i  =  suc  j )
2518, 24bnj833 31585 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  =  suc  j )
26 simp2 984 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  n )
27133ad2ant3 1006 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  n  e.  D )
283bnj923 31595 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
29 elnn 6485 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
3028, 29sylan2 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  n  /\  n  e.  D )  ->  i  e.  om )
3126, 27, 30syl2anc 656 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  om )
3218, 31bnj832 31584 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  om )
33 vex 2973 . . . . . . . . . . . . . . . . . . . 20  |-  j  e. 
_V
3433bnj216 31557 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  suc  j  -> 
j  e.  i )
35 elnn 6485 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  i  /\  i  e.  om )  ->  j  e.  om )
3634, 35sylan 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =  suc  j  /\  i  e.  om )  ->  j  e.  om )
3725, 32, 36syl2anc 656 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  j  e.  om )
3818, 26bnj832 31584 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  n
)
3925, 38eqeltrrd 2516 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  suc  j  e.  n
)
402bnj589 31736 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4140biimpi 194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps 
->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4241bnj708 31582 . . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . . . 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 195 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
46453ad2ant3 1006 . . . . . . . . . . . . . . . . . 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 31584 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
4837, 39, 47mp2d 45 . . . . . . . . . . . . . . . 16  |-  ( th 
->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
49 fveq2 5688 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
5049eqeq1d 2449 . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 15  |-  ( th 
->  ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
5323, 52bnj1262 31638 . . . . . . . . . . . . . 14  |-  ( th 
->  ( f `  i
)  C_  A )
5421, 53bnj1023 31608 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
f `  i )  C_  A )
55 3anass 964 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  <->  ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) ) )
5655imbi1i 325 . . . . . . . . . . . . . 14  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5756exbii 1639 . . . . . . . . . . . . 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 208 . . . . . . . . . . . 12  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch )
)  ->  ( f `  i )  C_  A
)
591biimpi 194 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
607, 59bnj771 31591 . . . . . . . . . . . . . 14  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
61 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
62 bnj213 31709 . . . . . . . . . . . . . . . 16  |-  pred ( X ,  A ,  R )  C_  A
63 sseq1 3374 . . . . . . . . . . . . . . . 16  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( ( f `  (/) )  C_  A  <->  pred ( X ,  A ,  R
)  C_  A )
)
6462, 63mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  C_  A )
65 sseq1 3374 . . . . . . . . . . . . . . . 16  |-  ( ( f `  i )  =  ( f `  (/) )  ->  ( (
f `  i )  C_  A  <->  ( f `  (/) )  C_  A )
)
6665biimpar 482 . . . . . . . . . . . . . . 15  |-  ( ( ( f `  i
)  =  ( f `
 (/) )  /\  (
f `  (/) )  C_  A )  ->  (
f `  i )  C_  A )
6761, 64, 66syl2an 474 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  C_  A
)
6860, 67sylan2 471 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  ch )  ->  ( f `  i )  C_  A
)
6968adantrl 710 . . . . . . . . . . . 12  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A )
7058, 69bnj1109 31614 . . . . . . . . . . 11  |-  E. j
( ( i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )
71 19.9v 1722 . . . . . . . . . . 11  |-  ( E. j ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
)  <->  ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
) )
7270, 71mpbi 208 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ch )  ->  ( f `
 i )  C_  A )
7372expcom 435 . . . . . . . . 9  |-  ( ch 
->  ( i  e.  n  ->  ( f `  i
)  C_  A )
)
74 fndm 5507 . . . . . . . . . . 11  |-  ( f  Fn  n  ->  dom  f  =  n )
757, 74bnj770 31590 . . . . . . . . . 10  |-  ( ch 
->  dom  f  =  n )
76 eleq2 2502 . . . . . . . . . . 11  |-  ( dom  f  =  n  -> 
( i  e.  dom  f 
<->  i  e.  n ) )
7776imbi1d 317 . . . . . . . . . 10  |-  ( dom  f  =  n  -> 
( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7875, 77syl 16 . . . . . . . . 9  |-  ( ch 
->  ( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7973, 78mpbird 232 . . . . . . . 8  |-  ( ch 
->  ( i  e.  dom  f  ->  ( f `  i )  C_  A
) )
8011, 79ralrimi 2795 . . . . . . 7  |-  ( ch 
->  A. i  e.  dom  f ( f `  i )  C_  A
)
8180exlimiv 1693 . . . . . 6  |-  ( E. n ch  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
828, 81sylbi 195 . . . . 5  |-  ( f  e.  B  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
83 ss2iun 4183 . . . . . 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 31618 . . . . . 6  |-  U_ i  e.  dom  f  A  C_  A
8583, 84syl6ss 3365 . . . . 5  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
8682, 85syl 16 . . . 4  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
876, 86mprg 2783 . . 3  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  U_ f  e.  B  A
884bnj1317 31649 . . . 4  |-  ( w  e.  B  ->  A. f  w  e.  B )
8988bnj1146 31619 . . 3  |-  U_ f  e.  B  A  C_  A
9087, 89sstri 3362 . 2  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  A
915, 90eqsstri 3383 1  |-  trCl ( X ,  A ,  R )  C_  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   E.wrex 2714    \ cdif 3322    C_ wss 3325   (/)c0 3634   {csn 3874   U_ciun 4168   suc csuc 4717   dom cdm 4836    Fn wfn 5410   ` cfv 5415   omcom 6475    /\ w-bnj17 31508    predc-bnj14 31510    trClc-bnj18 31516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-iota 5378  df-fn 5418  df-fv 5423  df-om 6476  df-bnj17 31509  df-bnj14 31511  df-bnj18 31517
This theorem is referenced by:  bnj1147  31819
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