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Theorem bnj1145 33777
Description: Technical lemma for bnj69 33794. 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 33712 . 2  |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f ( f `
 i )
6 ss2iun 4331 . . . 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 33762 . . . . . 6  |-  ( f  e.  B  <->  E. n ch )
92bnj1095 33568 . . . . . . . . 9  |-  ( ps 
->  A. i ps )
109, 7bnj1096 33569 . . . . . . . 8  |-  ( ch 
->  A. i ch )
113bnj1098 33570 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
127bnj1232 33590 . . . . . . . . . . . . . . . . . 18  |-  ( ch 
->  n  e.  D
)
13123anim3i 1185 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D ) )
1411, 13bnj1101 33571 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
j  e.  n  /\  i  =  suc  j ) )
15 ancl 546 . . . . . . . . . . . . . . . 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 33504 . . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )  <->  ( (
i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
1918exbii 1654 . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  th )
21 bnj213 33668 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
2221bnj226 33517 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
23 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  i  =  suc  j )  ->  i  =  suc  j )
2417, 23bnj833 33544 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  =  suc  j )
25 simp2 998 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  n )
26123ad2ant3 1020 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  n  e.  D )
273bnj923 33554 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  D  ->  n  e.  om )
28 elnn 6695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
2927, 28sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( i  e.  n  /\  n  e.  D )  ->  i  e.  om )
3025, 26, 29syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  i  e.  om )
3117, 30bnj832 33543 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  om )
32 vex 3098 . . . . . . . . . . . . . . . . . . . 20  |-  j  e. 
_V
3332bnj216 33515 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  suc  j  -> 
j  e.  i )
34 elnn 6695 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  i  /\  i  e.  om )  ->  j  e.  om )
3533, 34sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =  suc  j  /\  i  e.  om )  ->  j  e.  om )
3624, 31, 35syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  j  e.  om )
3717, 25bnj832 33543 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  i  e.  n
)
3824, 37eqeltrrd 2532 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  suc  j  e.  n
)
392bnj589 33695 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4039biimpi 194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps 
->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
4140bnj708 33541 . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . 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 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 ) ) ) )
447, 43sylbi 195 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
45443ad2ant3 1020 . . . . . . . . . . . . . . . . . 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 33543 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
4736, 38, 46mp2d 45 . . . . . . . . . . . . . . . 16  |-  ( th 
->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )
48 fveq2 5856 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
4948eqeq1d 2445 . . . . . . . . . . . . . . . . 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 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 ) ) )
5147, 50mpbird 232 . . . . . . . . . . . . . . 15  |-  ( th 
->  ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
5222, 51bnj1262 33597 . . . . . . . . . . . . . 14  |-  ( th 
->  ( f `  i
)  C_  A )
5320, 52bnj1023 33567 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  (
f `  i )  C_  A )
54 3anass 978 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  <->  ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) ) )
5554imbi1i 325 . . . . . . . . . . . . . 14  |-  ( ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )  <->  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A ) )
5655exbii 1654 . . . . . . . . . . . . 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 208 . . . . . . . . . . . 12  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ch )
)  ->  ( f `  i )  C_  A
)
581biimpi 194 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
597, 58bnj771 33550 . . . . . . . . . . . . . 14  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
60 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
61 bnj213 33668 . . . . . . . . . . . . . . . 16  |-  pred ( X ,  A ,  R )  C_  A
62 sseq1 3510 . . . . . . . . . . . . . . . 16  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( ( f `  (/) )  C_  A  <->  pred ( X ,  A ,  R
)  C_  A )
)
6361, 62mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  ->  ( f `  (/) )  C_  A )
64 sseq1 3510 . . . . . . . . . . . . . . . 16  |-  ( ( f `  i )  =  ( f `  (/) )  ->  ( (
f `  i )  C_  A  <->  ( f `  (/) )  C_  A )
)
6564biimpar 485 . . . . . . . . . . . . . . 15  |-  ( ( ( f `  i
)  =  ( f `
 (/) )  /\  (
f `  (/) )  C_  A )  ->  (
f `  i )  C_  A )
6660, 63, 65syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  C_  A
)
6759, 66sylan2 474 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  ch )  ->  ( f `  i )  C_  A
)
6867adantrl 715 . . . . . . . . . . . 12  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ch ) )  ->  (
f `  i )  C_  A )
6957, 68bnj1109 33573 . . . . . . . . . . 11  |-  E. j
( ( i  e.  n  /\  ch )  ->  ( f `  i
)  C_  A )
70 19.9v 1741 . . . . . . . . . . 11  |-  ( E. j ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
)  <->  ( ( i  e.  n  /\  ch )  ->  ( f `  i )  C_  A
) )
7169, 70mpbi 208 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ch )  ->  ( f `
 i )  C_  A )
7271expcom 435 . . . . . . . . 9  |-  ( ch 
->  ( i  e.  n  ->  ( f `  i
)  C_  A )
)
73 fndm 5670 . . . . . . . . . . 11  |-  ( f  Fn  n  ->  dom  f  =  n )
747, 73bnj770 33549 . . . . . . . . . 10  |-  ( ch 
->  dom  f  =  n )
75 eleq2 2516 . . . . . . . . . . 11  |-  ( dom  f  =  n  -> 
( i  e.  dom  f 
<->  i  e.  n ) )
7675imbi1d 317 . . . . . . . . . 10  |-  ( dom  f  =  n  -> 
( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7774, 76syl 16 . . . . . . . . 9  |-  ( ch 
->  ( ( i  e. 
dom  f  ->  (
f `  i )  C_  A )  <->  ( i  e.  n  ->  ( f `
 i )  C_  A ) ) )
7872, 77mpbird 232 . . . . . . . 8  |-  ( ch 
->  ( i  e.  dom  f  ->  ( f `  i )  C_  A
) )
7910, 78hbralrimi 2839 . . . . . . 7  |-  ( ch 
->  A. i  e.  dom  f ( f `  i )  C_  A
)
8079exlimiv 1709 . . . . . 6  |-  ( E. n ch  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
818, 80sylbi 195 . . . . 5  |-  ( f  e.  B  ->  A. i  e.  dom  f ( f `
 i )  C_  A )
82 ss2iun 4331 . . . . . 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 33577 . . . . . 6  |-  U_ i  e.  dom  f  A  C_  A
8482, 83syl6ss 3501 . . . . 5  |-  ( A. i  e.  dom  f ( f `  i ) 
C_  A  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
8581, 84syl 16 . . . 4  |-  ( f  e.  B  ->  U_ i  e.  dom  f ( f `
 i )  C_  A )
866, 85mprg 2806 . . 3  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  U_ f  e.  B  A
874bnj1317 33608 . . . 4  |-  ( w  e.  B  ->  A. f  w  e.  B )
8887bnj1146 33578 . . 3  |-  U_ f  e.  B  A  C_  A
8986, 88sstri 3498 . 2  |-  U_ f  e.  B  U_ i  e. 
dom  f ( f `
 i )  C_  A
905, 89eqsstri 3519 1  |-  trCl ( X ,  A ,  R )  C_  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428    =/= wne 2638   A.wral 2793   E.wrex 2794    \ cdif 3458    C_ wss 3461   (/)c0 3770   {csn 4014   U_ciun 4315   suc csuc 4870   dom cdm 4989    Fn wfn 5573   ` cfv 5578   omcom 6685    /\ w-bnj17 33466    predc-bnj14 33468    trClc-bnj18 33474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-iota 5541  df-fn 5581  df-fv 5586  df-om 6686  df-bnj17 33467  df-bnj14 33469  df-bnj18 33475
This theorem is referenced by:  bnj1147  33778
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