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Theorem fin23lem31 7969
Description: Lemma for fin23 8015. 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 7956 . . 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 7967 . . . . 5  |-  U. ran  Z 
C_  U. ran  t
98a1i 10 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C_  U. ran  t )
103, 1fin23lem21 7965 . . . . . . 7  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  =/=  (/) )
1110ancoms 439 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  |^| ran  U  =/=  (/) )
12 n0 3464 . . . . . 6  |-  ( |^| ran 
U  =/=  (/)  <->  E. a 
a  e.  |^| ran  U )
1311, 12sylib 188 . . . . 5  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  E. a 
a  e.  |^| ran  U )
143fnseqom 6467 . . . . . . . . . . . . . . . 16  |-  U  Fn  om
15 fndm 5343 . . . . . . . . . . . . . . . 16  |-  ( U  Fn  om  ->  dom  U  =  om )
1614, 15ax-mp 8 . . . . . . . . . . . . . . 15  |-  dom  U  =  om
17 peano1 4675 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
18 ne0i 3461 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1917, 18ax-mp 8 . . . . . . . . . . . . . . 15  |-  om  =/=  (/)
2016, 19eqnetri 2463 . . . . . . . . . . . . . 14  |-  dom  U  =/=  (/)
21 dm0rn0 4895 . . . . . . . . . . . . . . 15  |-  ( dom 
U  =  (/)  <->  ran  U  =  (/) )
2221necon3bii 2478 . . . . . . . . . . . . . 14  |-  ( dom 
U  =/=  (/)  <->  ran  U  =/=  (/) )
2320, 22mpbi 199 . . . . . . . . . . . . 13  |-  ran  U  =/=  (/)
24 intssuni 3884 . . . . . . . . . . . . 13  |-  ( ran 
U  =/=  (/)  ->  |^| ran  U 
C_  U. ran  U )
2523, 24ax-mp 8 . . . . . . . . . . . 12  |-  |^| ran  U 
C_  U. ran  U
263fin23lem16 7961 . . . . . . . . . . . 12  |-  U. ran  U  =  U. ran  t
2725, 26sseqtri 3210 . . . . . . . . . . 11  |-  |^| ran  U 
C_  U. ran  t
2827sseli 3176 . . . . . . . . . 10  |-  ( a  e.  |^| ran  U  -> 
a  e.  U. ran  t )
2928adantl 452 . . . . . . . . 9  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  a  e.  U. ran  t )
30 f1fun 5439 . . . . . . . . . . . . . . 15  |-  ( t : om -1-1-> V  ->  Fun  t )
3130adantr 451 . . . . . . . . . . . . . 14  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  Fun  t )
323, 1, 4, 5, 6, 7fin23lem30 7968 . . . . . . . . . . . . . 14  |-  ( Fun  t  ->  ( U. ran  Z  i^i  |^| ran  U )  =  (/) )
3331, 32syl 15 . . . . . . . . . . . . 13  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  ( U. ran  Z  i^i  |^| ran 
U )  =  (/) )
34 disj 3495 . . . . . . . . . . . . 13  |-  ( ( U. ran  Z  i^i  |^|
ran  U )  =  (/) 
<-> 
A. a  e.  U. ran  Z  -.  a  e. 
|^| ran  U )
3533, 34sylib 188 . . . . . . . . . . . 12  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U
)
36 rsp 2603 . . . . . . . . . . . 12  |-  ( A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U  ->  ( a  e. 
U. ran  Z  ->  -.  a  e.  |^| ran  U ) )
3735, 36syl 15 . . . . . . . . . . 11  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  U. ran  Z  ->  -.  a  e.  |^|
ran  U ) )
3837con2d 107 . . . . . . . . . 10  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  |^| ran  U  ->  -.  a  e.  U.
ran  Z ) )
3938imp 418 . . . . . . . . 9  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  -.  a  e.  U.
ran  Z )
40 nelne1 2535 . . . . . . . . 9  |-  ( ( a  e.  U. ran  t  /\  -.  a  e. 
U. ran  Z )  ->  U. ran  t  =/=  U. ran  Z )
4129, 39, 40syl2anc 642 . . . . . . . 8  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  t  =/=  U. ran  Z )
4241necomd 2529 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  Z  =/=  U. ran  t )
4342ex 423 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  |^| ran  U  ->  U. ran  Z  =/=  U. ran  t ) )
4443exlimdv 1664 . . . . 5  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  ( E. a  a  e.  |^|
ran  U  ->  U. ran  Z  =/=  U. ran  t
) )
4513, 44mpd 14 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z  =/=  U. ran  t
)
46 df-pss 3168 . . . 4  |-  ( U. ran  Z  C.  U. ran  t 
<->  ( U. ran  Z  C_ 
U. ran  t  /\  U.
ran  Z  =/=  U. ran  t ) )
479, 45, 46sylanbrc 645 . . 3  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C.  U. ran  t )
482, 47sylan2 460 . 2  |-  ( ( t : om -1-1-> V  /\  ( G  e.  F  /\  U. ran  t  C_  G ) )  ->  U. ran  Z  C.  U. ran  t )
49483impb 1147 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   ifcif 3565   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   class class class wbr 4023    e. cmpt 4077   suc csuc 4394   omcom 4656   dom cdm 4689   ran crn 4690    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297  seq𝜔cseqom 6459    ^m cmap 6772    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  fin23lem32  7970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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