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Theorem bnj1145 29798
Description: Technical lemma for bnj69 29815. 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 29733 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
6 ss2iun 4312 . . . 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 29783 . . . . . 6  |-  ( f  e.  B  <->  E. n ch )
92bnj1095 29589 . . . . . . . . 9  |-  ( ps 
->  A. i ps )
109, 7bnj1096 29590 . . . . . . . 8  |-  ( ch 
->  A. i ch )
113bnj1098 29591 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
127bnj1232 29611 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  n  e.  D
)
13123anim3i 1193 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
1411, 13bnj1101 29592 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
j  e.  n  /\  i  =  suc  j ) )
15 ancl 548 . . . . . . . . . . . . . . . 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 ) ) ) )
1614, 15bnj101 29525 . . . . . . . . . . . . . . 15  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
17 bnj1145.6 . . . . . . . . . . . . . . . . 17  |-  ( th  <->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
1817imbi2i 313 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )  <->  ( (
i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
1918exbii 1712 . . . . . . . . . . . . . . 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 ) ) ) )
2016, 19mpbir 212 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )
21 bnj213 29689 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
2221bnj226 29538 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
23 simpr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  i  =  suc  j )
2417, 23bnj833 29565 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  =  suc  j )
25 simp2 1006 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  n )
26123ad2ant3 1028 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  n  e.  D )
273bnj923 29575 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
28 elnn 6713 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
2927, 28sylan2 476 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  n  /\  n  e.  D )  ->  i  e.  om )
3025, 26, 29syl2anc 665 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  om )
3117, 30bnj832 29564 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  om )
32 vex 3084 . . . . . . . . . . . . . . . . . . . 20  |-  j  e. 
_V
3332bnj216 29536 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  suc  j  -> 
j  e.  i )
34 elnn 6713 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  i  /\  i  e.  om )  ->  j  e.  om )
3533, 34sylan 473 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =  suc  j  /\  i  e.  om )  ->  j  e.  om )
3624, 31, 35syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  j  e.  om )
3717, 25bnj832 29564 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  n
)
3824, 37eqeltrrd 2511 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  suc  j  e.  n
)
392bnj589 29716 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4039biimpi 197 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps 
->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4140bnj708 29562 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
42 rsp 2791 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
447, 43sylbi 198 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
45443ad2ant3 1028 . . . . . . . . . . . . . . . . . 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 ) ) ) )
4617, 45bnj832 29564 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
4736, 38, 46mp2d 46 . . . . . . . . . . . . . . . 16  |-  ( th 
->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
48 fveq2 5878 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
4948eqeq1d 2424 . . . . . . . . . . . . . . . . 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 ) ) )
5024, 49syl 17 . . . . . . . . . . . . . . . 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 ) ) )
5147, 50mpbird 235 . . . . . . . . . . . . . . 15  |-  ( th 
->  ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
5222, 51bnj1262 29618 . . . . . . . . . . . . . 14  |-  ( th 
->  ( f `  i
)  C_  A )
5320, 52bnj1023 29588 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
f `  i )  C_  A )
54 3anass 986 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  <->  ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) ) )
5554imbi1i 326 . . . . . . . . . . . . . 14  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5655exbii 1712 . . . . . . . . . . . . 13  |-  ( E. j ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  E. j ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5753, 56mpbi 211 . . . . . . . . . . . 12  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch )
)  ->  ( f `  i )  C_  A
)
581biimpi 197 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
597, 58bnj771 29571 . . . . . . . . . . . . . 14  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
60 fveq2 5878 . . . . . . . . . . . . . . 15  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
61 bnj213 29689 . . . . . . . . . . . . . . . 16  |-  pred ( X ,  A ,  R )  C_  A
62 sseq1 3485 . . . . . . . . . . . . . . . 16  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( ( f `  (/) )  C_  A  <->  pred ( X ,  A ,  R
)  C_  A )
)
6361, 62mpbiri 236 . . . . . . . . . . . . . . 15  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  C_  A )
64 sseq1 3485 . . . . . . . . . . . . . . . 16  |-  ( ( f `  i )  =  ( f `  (/) )  ->  ( (
f `  i )  C_  A  <->  ( f `  (/) )  C_  A )
)
6564biimpar 487 . . . . . . . . . . . . . . 15  |-  ( ( ( f `  i
)  =  ( f `
 (/) )  /\  (
f `  (/) )  C_  A )  ->  (
f `  i )  C_  A )
6660, 63, 65syl2an 479 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  C_  A
)
6759, 66sylan2 476 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  ch )  ->  ( f `  i )  C_  A
)
6867adantrl 720 . . . . . . . . . . . 12  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A )
6957, 68bnj1109 29594 . . . . . . . . . . 11  |-  E. j
( ( i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )
70 19.9v 1801 . . . . . . . . . . 11  |-  ( E. j ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
)  <->  ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
) )
7169, 70mpbi 211 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ch )  ->  ( f `
 i )  C_  A )
7271expcom 436 . . . . . . . . 9  |-  ( ch 
->  ( i  e.  n  ->  ( f `  i
)  C_  A )
)
73 fndm 5690 . . . . . . . . . . 11  |-  ( f  Fn  n  ->  dom  f  =  n )
747, 73bnj770 29570 . . . . . . . . . 10  |-  ( ch 
->  dom  f  =  n )
75 eleq2 2495 . . . . . . . . . . 11  |-  ( dom  f  =  n  -> 
( i  e.  dom  f 
<->  i  e.  n ) )
7675imbi1d 318 . . . . . . . . . 10  |-  ( dom  f  =  n  -> 
( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7774, 76syl 17 . . . . . . . . 9  |-  ( ch 
->  ( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7872, 77mpbird 235 . . . . . . . 8  |-  ( ch 
->  ( i  e.  dom  f  ->  ( f `  i )  C_  A
) )
7910, 78hbralrimi 2821 . . . . . . 7  |-  ( ch 
->  A. i  e.  dom  f ( f `  i )  C_  A
)
8079exlimiv 1766 . . . . . 6  |-  ( E. n ch  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
818, 80sylbi 198 . . . . 5  |-  ( f  e.  B  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
82 ss2iun 4312 . . . . . 6  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  U_ i  e.  dom  f  A )
83 bnj1143 29598 . . . . . 6  |-  U_ i  e.  dom  f  A  C_  A
8482, 83syl6ss 3476 . . . . 5  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
8581, 84syl 17 . . . 4  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
866, 85mprg 2788 . . 3  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  U_ f  e.  B  A
874bnj1317 29629 . . . 4  |-  ( w  e.  B  ->  A. f  w  e.  B )
8887bnj1146 29599 . . 3  |-  U_ f  e.  B  A  C_  A
8986, 88sstri 3473 . 2  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  A
905, 89eqsstri 3494 1  |-  trCl ( X ,  A ,  R )  C_  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1868   {cab 2407    =/= wne 2618   A.wral 2775   E.wrex 2776    \ cdif 3433    C_ wss 3436   (/)c0 3761   {csn 3996   U_ciun 4296   dom cdm 4850   suc csuc 5441    Fn wfn 5593   ` cfv 5598   omcom 6703    /\ w-bnj17 29487    predc-bnj14 29489    trClc-bnj18 29495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4761  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fn 5601  df-fv 5606  df-om 6704  df-bnj17 29488  df-bnj14 29490  df-bnj18 29496
This theorem is referenced by:  bnj1147  29799
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