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Theorem fin23lem31 8740
Description: Lemma for fin23 8786. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
fin23lem17.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
fin23lem.b  |-  P  =  { v  e.  om  |  |^| ran  U  C_  ( t `  v
) }
fin23lem.c  |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P  ( x  i^i  P ) 
~~  w ) )
fin23lem.d  |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) ) 
~~  w ) )
fin23lem.e  |-  Z  =  if ( P  e. 
Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q ) )
Assertion
Ref Expression
fin23lem31  |-  ( ( t : om -1-1-> V  /\  G  e.  F  /\  U. ran  t  C_  G )  ->  U. ran  Z 
C.  U. ran  t )
Distinct variable groups:    g, i,
t, u, v, x, z, a    F, a, t    V, a    w, a, x, z, P    v,
a, R, i, u    U, a, i, u, v, z    Z, a    g, a, G, t, x
Allowed substitution hints:    P( v, u, t, g, i)    Q( x, z, w, v, u, t, g, i, a)    R( x, z, w, t, g)    U( x, w, t, g)    F( x, z, w, v, u, g, i)    G( z, w, v, u, i)    V( x, z, w, v, u, t, g, i)    Z( x, z, w, v, u, t, g, i)

Proof of Theorem fin23lem31
StepHypRef Expression
1 fin23lem17.f . . . 4  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
21ssfin3ds 8727 . . 3  |-  ( ( G  e.  F  /\  U.
ran  t  C_  G
)  ->  U. ran  t  e.  F )
3 fin23lem.a . . . . . 6  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
4 fin23lem.b . . . . . 6  |-  P  =  { v  e.  om  |  |^| ran  U  C_  ( t `  v
) }
5 fin23lem.c . . . . . 6  |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P  ( x  i^i  P ) 
~~  w ) )
6 fin23lem.d . . . . . 6  |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) ) 
~~  w ) )
7 fin23lem.e . . . . . 6  |-  Z  =  if ( P  e. 
Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q ) )
83, 1, 4, 5, 6, 7fin23lem29 8738 . . . . 5  |-  U. ran  Z 
C_  U. ran  t
98a1i 11 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C_  U. ran  t )
103, 1fin23lem21 8736 . . . . . . 7  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  =/=  (/) )
1110ancoms 453 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  |^| ran  U  =/=  (/) )
12 n0 3803 . . . . . 6  |-  ( |^| ran 
U  =/=  (/)  <->  E. a 
a  e.  |^| ran  U )
1311, 12sylib 196 . . . . 5  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  E. a 
a  e.  |^| ran  U )
143fnseqom 7138 . . . . . . . . . . . . . 14  |-  U  Fn  om
15 fndm 5686 . . . . . . . . . . . . . 14  |-  ( U  Fn  om  ->  dom  U  =  om )
1614, 15ax-mp 5 . . . . . . . . . . . . 13  |-  dom  U  =  om
17 peano1 6718 . . . . . . . . . . . . . 14  |-  (/)  e.  om
1817ne0ii 3800 . . . . . . . . . . . . 13  |-  om  =/=  (/)
1916, 18eqnetri 2753 . . . . . . . . . . . 12  |-  dom  U  =/=  (/)
20 dm0rn0 5229 . . . . . . . . . . . . 13  |-  ( dom 
U  =  (/)  <->  ran  U  =  (/) )
2120necon3bii 2725 . . . . . . . . . . . 12  |-  ( dom 
U  =/=  (/)  <->  ran  U  =/=  (/) )
2219, 21mpbi 208 . . . . . . . . . . 11  |-  ran  U  =/=  (/)
23 intssuni 4311 . . . . . . . . . . 11  |-  ( ran 
U  =/=  (/)  ->  |^| ran  U 
C_  U. ran  U )
2422, 23ax-mp 5 . . . . . . . . . 10  |-  |^| ran  U 
C_  U. ran  U
253fin23lem16 8732 . . . . . . . . . 10  |-  U. ran  U  =  U. ran  t
2624, 25sseqtri 3531 . . . . . . . . 9  |-  |^| ran  U 
C_  U. ran  t
2726sseli 3495 . . . . . . . 8  |-  ( a  e.  |^| ran  U  -> 
a  e.  U. ran  t )
2827adantl 466 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  a  e.  U. ran  t )
29 f1fun 5789 . . . . . . . . . . . . 13  |-  ( t : om -1-1-> V  ->  Fun  t )
3029adantr 465 . . . . . . . . . . . 12  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  Fun  t )
313, 1, 4, 5, 6, 7fin23lem30 8739 . . . . . . . . . . . 12  |-  ( Fun  t  ->  ( U. ran  Z  i^i  |^| ran  U )  =  (/) )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  ( U. ran  Z  i^i  |^| ran 
U )  =  (/) )
33 disj 3870 . . . . . . . . . . 11  |-  ( ( U. ran  Z  i^i  |^|
ran  U )  =  (/) 
<-> 
A. a  e.  U. ran  Z  -.  a  e. 
|^| ran  U )
3432, 33sylib 196 . . . . . . . . . 10  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U
)
35 rsp 2823 . . . . . . . . . 10  |-  ( A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U  ->  ( a  e. 
U. ran  Z  ->  -.  a  e.  |^| ran  U ) )
3634, 35syl 16 . . . . . . . . 9  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  U. ran  Z  ->  -.  a  e.  |^|
ran  U ) )
3736con2d 115 . . . . . . . 8  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  |^| ran  U  ->  -.  a  e.  U.
ran  Z ) )
3837imp 429 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  -.  a  e.  U.
ran  Z )
39 nelne1 2786 . . . . . . 7  |-  ( ( a  e.  U. ran  t  /\  -.  a  e. 
U. ran  Z )  ->  U. ran  t  =/=  U. ran  Z )
4028, 38, 39syl2anc 661 . . . . . 6  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  t  =/=  U. ran  Z )
4140necomd 2728 . . . . 5  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  Z  =/=  U. ran  t )
4213, 41exlimddv 1727 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z  =/=  U. ran  t
)
43 df-pss 3487 . . . 4  |-  ( U. ran  Z  C.  U. ran  t  <->  ( U. ran  Z  C_  U.
ran  t  /\  U. ran  Z  =/=  U. ran  t ) )
449, 42, 43sylanbrc 664 . . 3  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C.  U. ran  t )
452, 44sylan2 474 . 2  |-  ( ( t : om -1-1-> V  /\  ( G  e.  F  /\  U. ran  t  C_  G ) )  ->  U. ran  Z  C.  U. ran  t )
46453impb 1192 1  |-  ( ( t : om -1-1-> V  /\  G  e.  F  /\  U. ran  t  C_  G )  ->  U. ran  Z 
C.  U. ran  t )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471    C. wpss 3472   (/)c0 3793   ifcif 3944   ~Pcpw 4015   U.cuni 4251   |^|cint 4288   class class class wbr 4456    |-> cmpt 4515   suc csuc 4889   dom cdm 5008   ran crn 5009    o. ccom 5012   Fun wfun 5588    Fn wfn 5589   -1-1->wf1 5591   ` cfv 5594   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130    ^m cmap 7438    ~~ cen 7532   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337
This theorem is referenced by:  fin23lem32  8741
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