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Theorem fin23lem31 8770
Description: Lemma for fin23 8816. 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 8757 . . 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 8768 . . . . 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 8766 . . . . . . 7  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  =/=  (/) )
1110ancoms 455 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  |^| ran  U  =/=  (/) )
12 n0 3740 . . . . . 6  |-  ( |^| ran 
U  =/=  (/)  <->  E. a 
a  e.  |^| ran  U )
1311, 12sylib 200 . . . . 5  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  E. a 
a  e.  |^| ran  U )
143fnseqom 7169 . . . . . . . . . . . . . 14  |-  U  Fn  om
15 fndm 5673 . . . . . . . . . . . . . 14  |-  ( U  Fn  om  ->  dom  U  =  om )
1614, 15ax-mp 5 . . . . . . . . . . . . 13  |-  dom  U  =  om
17 peano1 6709 . . . . . . . . . . . . . 14  |-  (/)  e.  om
1817ne0ii 3737 . . . . . . . . . . . . 13  |-  om  =/=  (/)
1916, 18eqnetri 2693 . . . . . . . . . . . 12  |-  dom  U  =/=  (/)
20 dm0rn0 5050 . . . . . . . . . . . . 13  |-  ( dom 
U  =  (/)  <->  ran  U  =  (/) )
2120necon3bii 2675 . . . . . . . . . . . 12  |-  ( dom 
U  =/=  (/)  <->  ran  U  =/=  (/) )
2219, 21mpbi 212 . . . . . . . . . . 11  |-  ran  U  =/=  (/)
23 intssuni 4256 . . . . . . . . . . 11  |-  ( ran 
U  =/=  (/)  ->  |^| ran  U 
C_  U. ran  U )
2422, 23ax-mp 5 . . . . . . . . . 10  |-  |^| ran  U 
C_  U. ran  U
253fin23lem16 8762 . . . . . . . . . 10  |-  U. ran  U  =  U. ran  t
2624, 25sseqtri 3463 . . . . . . . . 9  |-  |^| ran  U 
C_  U. ran  t
2726sseli 3427 . . . . . . . 8  |-  ( a  e.  |^| ran  U  -> 
a  e.  U. ran  t )
2827adantl 468 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  a  e.  U. ran  t )
29 f1fun 5779 . . . . . . . . . . . . 13  |-  ( t : om -1-1-> V  ->  Fun  t )
3029adantr 467 . . . . . . . . . . . 12  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  Fun  t )
313, 1, 4, 5, 6, 7fin23lem30 8769 . . . . . . . . . . . 12  |-  ( Fun  t  ->  ( U. ran  Z  i^i  |^| ran  U )  =  (/) )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  ( U. ran  Z  i^i  |^| ran 
U )  =  (/) )
33 disj 3804 . . . . . . . . . . 11  |-  ( ( U. ran  Z  i^i  |^|
ran  U )  =  (/) 
<-> 
A. a  e.  U. ran  Z  -.  a  e. 
|^| ran  U )
3432, 33sylib 200 . . . . . . . . . 10  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U
)
35 rsp 2753 . . . . . . . . . 10  |-  ( A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U  ->  ( a  e. 
U. ran  Z  ->  -.  a  e.  |^| ran  U ) )
3634, 35syl 17 . . . . . . . . 9  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  U. ran  Z  ->  -.  a  e.  |^|
ran  U ) )
3736con2d 119 . . . . . . . 8  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  |^| ran  U  ->  -.  a  e.  U.
ran  Z ) )
3837imp 431 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  -.  a  e.  U.
ran  Z )
39 nelne1 2719 . . . . . . 7  |-  ( ( a  e.  U. ran  t  /\  -.  a  e. 
U. ran  Z )  ->  U. ran  t  =/=  U. ran  Z )
4028, 38, 39syl2anc 666 . . . . . 6  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  t  =/=  U. ran  Z )
4140necomd 2678 . . . . 5  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  Z  =/=  U. ran  t )
4213, 41exlimddv 1780 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z  =/=  U. ran  t
)
43 df-pss 3419 . . . 4  |-  ( U. ran  Z  C.  U. ran  t  <->  ( U. ran  Z  C_  U.
ran  t  /\  U. ran  Z  =/=  U. ran  t ) )
449, 42, 43sylanbrc 669 . . 3  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C.  U. ran  t )
452, 44sylan2 477 . 2  |-  ( ( t : om -1-1-> V  /\  ( G  e.  F  /\  U. ran  t  C_  G ) )  ->  U. ran  Z  C.  U. ran  t )
46453impb 1203 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 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   {cab 2436    =/= wne 2621   A.wral 2736   {crab 2740   _Vcvv 3044    \ cdif 3400    i^i cin 3402    C_ wss 3403    C. wpss 3404   (/)c0 3730   ifcif 3880   ~Pcpw 3950   U.cuni 4197   |^|cint 4233   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833   ran crn 4834    o. ccom 4837   suc csuc 5424   Fun wfun 5575    Fn wfn 5576   -1-1->wf1 5578   ` cfv 5581   iota_crio 6249  (class class class)co 6288    |-> cmpt2 6290   omcom 6689  seq𝜔cseqom 7161    ^m cmap 7469    ~~ cen 7563   Fincfn 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-seqom 7162  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370
This theorem is referenced by:  fin23lem32  8771
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