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Theorem fin23lem31 8059
Description: Lemma for fin23 8105. 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 8046 . . 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 8057 . . . . 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 8055 . . . . . . 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 3540 . . . . . 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 6554 . . . . . . . . . . . . . . . 16  |-  U  Fn  om
15 fndm 5425 . . . . . . . . . . . . . . . 16  |-  ( U  Fn  om  ->  dom  U  =  om )
1614, 15ax-mp 8 . . . . . . . . . . . . . . 15  |-  dom  U  =  om
17 peano1 4757 . . . . . . . . . . . . . . . 16  |-  (/)  e.  om
18 ne0i 3537 . . . . . . . . . . . . . . . 16  |-  ( (/)  e.  om  ->  om  =/=  (/) )
1917, 18ax-mp 8 . . . . . . . . . . . . . . 15  |-  om  =/=  (/)
2016, 19eqnetri 2538 . . . . . . . . . . . . . 14  |-  dom  U  =/=  (/)
21 dm0rn0 4977 . . . . . . . . . . . . . . 15  |-  ( dom 
U  =  (/)  <->  ran  U  =  (/) )
2221necon3bii 2553 . . . . . . . . . . . . . 14  |-  ( dom 
U  =/=  (/)  <->  ran  U  =/=  (/) )
2320, 22mpbi 199 . . . . . . . . . . . . 13  |-  ran  U  =/=  (/)
24 intssuni 3965 . . . . . . . . . . . . 13  |-  ( ran 
U  =/=  (/)  ->  |^| ran  U 
C_  U. ran  U )
2523, 24ax-mp 8 . . . . . . . . . . . 12  |-  |^| ran  U 
C_  U. ran  U
263fin23lem16 8051 . . . . . . . . . . . 12  |-  U. ran  U  =  U. ran  t
2725, 26sseqtri 3286 . . . . . . . . . . 11  |-  |^| ran  U 
C_  U. ran  t
2827sseli 3252 . . . . . . . . . 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 5522 . . . . . . . . . . . . . . 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 8058 . . . . . . . . . . . . . 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 3571 . . . . . . . . . . . . 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 2679 . . . . . . . . . . . 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 2610 . . . . . . . . 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 2604 . . . . . . 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 1636 . . . . 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 3244 . . . 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 1541    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   A.wral 2619   {crab 2623   _Vcvv 2864    \ cdif 3225    i^i cin 3227    C_ wss 3228    C. wpss 3229   (/)c0 3531   ifcif 3641   ~Pcpw 3701   U.cuni 3908   |^|cint 3943   class class class wbr 4104    e. cmpt 4158   suc csuc 4476   omcom 4738   dom cdm 4771   ran crn 4772    o. ccom 4775   Fun wfun 5331    Fn wfn 5332   -1-1->wf1 5334   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   iota_crio 6384  seq𝜔cseqom 6546    ^m cmap 6860    ~~ cen 6948   Fincfn 6951
This theorem is referenced by:  fin23lem32  8060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-seqom 6547  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662
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