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Theorem fin23lem31 8517
Description: Lemma for fin23 8563. 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 8504 . . 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 8515 . . . . 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 8513 . . . . . . 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 3651 . . . . . 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 6915 . . . . . . . . . . . . . 14  |-  U  Fn  om
15 fndm 5515 . . . . . . . . . . . . . 14  |-  ( U  Fn  om  ->  dom  U  =  om )
1614, 15ax-mp 5 . . . . . . . . . . . . 13  |-  dom  U  =  om
17 peano1 6500 . . . . . . . . . . . . . 14  |-  (/)  e.  om
18 ne0i 3648 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1917, 18ax-mp 5 . . . . . . . . . . . . 13  |-  om  =/=  (/)
2016, 19eqnetri 2630 . . . . . . . . . . . 12  |-  dom  U  =/=  (/)
21 dm0rn0 5061 . . . . . . . . . . . . 13  |-  ( dom 
U  =  (/)  <->  ran  U  =  (/) )
2221necon3bii 2645 . . . . . . . . . . . 12  |-  ( dom 
U  =/=  (/)  <->  ran  U  =/=  (/) )
2320, 22mpbi 208 . . . . . . . . . . 11  |-  ran  U  =/=  (/)
24 intssuni 4155 . . . . . . . . . . 11  |-  ( ran 
U  =/=  (/)  ->  |^| ran  U 
C_  U. ran  U )
2523, 24ax-mp 5 . . . . . . . . . 10  |-  |^| ran  U 
C_  U. ran  U
263fin23lem16 8509 . . . . . . . . . 10  |-  U. ran  U  =  U. ran  t
2725, 26sseqtri 3393 . . . . . . . . 9  |-  |^| ran  U 
C_  U. ran  t
2827sseli 3357 . . . . . . . 8  |-  ( a  e.  |^| ran  U  -> 
a  e.  U. ran  t )
2928adantl 466 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  a  e.  U. ran  t )
30 f1fun 5613 . . . . . . . . . . . . 13  |-  ( t : om -1-1-> V  ->  Fun  t )
3130adantr 465 . . . . . . . . . . . 12  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  Fun  t )
323, 1, 4, 5, 6, 7fin23lem30 8516 . . . . . . . . . . . 12  |-  ( Fun  t  ->  ( U. ran  Z  i^i  |^| ran  U )  =  (/) )
3331, 32syl 16 . . . . . . . . . . 11  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  ( U. ran  Z  i^i  |^| ran 
U )  =  (/) )
34 disj 3724 . . . . . . . . . . 11  |-  ( ( U. ran  Z  i^i  |^|
ran  U )  =  (/) 
<-> 
A. a  e.  U. ran  Z  -.  a  e. 
|^| ran  U )
3533, 34sylib 196 . . . . . . . . . 10  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U
)
36 rsp 2781 . . . . . . . . . 10  |-  ( A. a  e.  U. ran  Z  -.  a  e.  |^| ran  U  ->  ( a  e. 
U. ran  Z  ->  -.  a  e.  |^| ran  U ) )
3735, 36syl 16 . . . . . . . . 9  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  U. ran  Z  ->  -.  a  e.  |^|
ran  U ) )
3837con2d 115 . . . . . . . 8  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  (
a  e.  |^| ran  U  ->  -.  a  e.  U.
ran  Z ) )
3938imp 429 . . . . . . 7  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  -.  a  e.  U.
ran  Z )
40 nelne1 2706 . . . . . . 7  |-  ( ( a  e.  U. ran  t  /\  -.  a  e. 
U. ran  Z )  ->  U. ran  t  =/=  U. ran  Z )
4129, 39, 40syl2anc 661 . . . . . 6  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  t  =/=  U. ran  Z )
4241necomd 2700 . . . . 5  |-  ( ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  /\  a  e.  |^| ran  U )  ->  U. ran  Z  =/=  U. ran  t )
4313, 42exlimddv 1692 . . . 4  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z  =/=  U. ran  t
)
44 df-pss 3349 . . . 4  |-  ( U. ran  Z  C.  U. ran  t  <->  ( U. ran  Z  C_  U.
ran  t  /\  U. ran  Z  =/=  U. ran  t ) )
459, 43, 44sylanbrc 664 . . 3  |-  ( ( t : om -1-1-> V  /\  U. ran  t  e.  F )  ->  U. ran  Z 
C.  U. ran  t )
462, 45sylan2 474 . 2  |-  ( ( t : om -1-1-> V  /\  ( G  e.  F  /\  U. ran  t  C_  G ) )  ->  U. ran  Z  C.  U. ran  t )
47463impb 1183 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977    \ cdif 3330    i^i cin 3332    C_ wss 3333    C. wpss 3334   (/)c0 3642   ifcif 3796   ~Pcpw 3865   U.cuni 4096   |^|cint 4133   class class class wbr 4297    e. cmpt 4355   suc csuc 4726   dom cdm 4845   ran crn 4846    o. ccom 4849   Fun wfun 5417    Fn wfn 5418   -1-1->wf1 5420   ` cfv 5423   iota_crio 6056  (class class class)co 6096    e. cmpt2 6098   omcom 6481  seq𝜔cseqom 6907    ^m cmap 7219    ~~ cen 7312   Fincfn 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-seqom 6908  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114
This theorem is referenced by:  fin23lem32  8518
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