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Theorem fin23lem17 8519
Description: Lemma for fin23 8570. By ? Fin3DS ? ,  U achieves its minimum ( X in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
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 ) }
Assertion
Ref Expression
fin23lem17  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
Distinct variable groups:    g, i,
t, u, x, a    F, a, t    V, a   
x, a    U, a,
i, u    g, a
Allowed substitution hints:    U( x, t, g)    F( x, u, g, i)    V( x, u, t, g, i)

Proof of Theorem fin23lem17
Dummy variables  c 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 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 )
21fnseqom 6922 . . . . 5  |-  U  Fn  om
3 dffn3 5578 . . . . 5  |-  ( U  Fn  om  <->  U : om
--> ran  U )
42, 3mpbi 208 . . . 4  |-  U : om
--> ran  U
5 pwuni 4535 . . . . 5  |-  ran  U  C_ 
~P U. ran  U
61fin23lem16 8516 . . . . . 6  |-  U. ran  U  =  U. ran  t
76pweqi 3876 . . . . 5  |-  ~P U. ran  U  =  ~P U. ran  t
85, 7sseqtri 3400 . . . 4  |-  ran  U  C_ 
~P U. ran  t
9 fss 5579 . . . 4  |-  ( ( U : om --> ran  U  /\  ran  U  C_  ~P U.
ran  t )  ->  U : om --> ~P U. ran  t )
104, 8, 9mp2an 672 . . 3  |-  U : om
--> ~P U. ran  t
11 vex 2987 . . . . . . 7  |-  t  e. 
_V
1211rnex 6524 . . . . . 6  |-  ran  t  e.  _V
1312uniex 6388 . . . . 5  |-  U. ran  t  e.  _V
1413pwex 4487 . . . 4  |-  ~P U. ran  t  e.  _V
15 f1f 5618 . . . . . 6  |-  ( t : om -1-1-> V  -> 
t : om --> V )
16 dmfex 6547 . . . . . 6  |-  ( ( t  e.  _V  /\  t : om --> V )  ->  om  e.  _V )
1711, 15, 16sylancr 663 . . . . 5  |-  ( t : om -1-1-> V  ->  om  e.  _V )
1817adantl 466 . . . 4  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  om  e.  _V )
19 elmapg 7239 . . . 4  |-  ( ( ~P U. ran  t  e.  _V  /\  om  e.  _V )  ->  ( U  e.  ( ~P U. ran  t  ^m  om )  <->  U : om --> ~P U. ran  t ) )
2014, 18, 19sylancr 663 . . 3  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  -> 
( U  e.  ( ~P U. ran  t  ^m  om )  <->  U : om
--> ~P U. ran  t
) )
2110, 20mpbiri 233 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  U  e.  ( ~P U.
ran  t  ^m  om ) )
22 fin23lem17.f . . . . 5  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
2322isfin3ds 8510 . . . 4  |-  ( U. ran  t  e.  F  ->  ( U. ran  t  e.  F  <->  A. b  e.  ( ~P U. ran  t  ^m  om ) ( A. c  e.  om  (
b `  suc  c ) 
C_  ( b `  c )  ->  |^| ran  b  e.  ran  b ) ) )
2423ibi 241 . . 3  |-  ( U. ran  t  e.  F  ->  A. b  e.  ( ~P U. ran  t  ^m  om ) ( A. c  e.  om  (
b `  suc  c ) 
C_  ( b `  c )  ->  |^| ran  b  e.  ran  b ) )
2524adantr 465 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  A. b  e.  ( ~P U. ran  t  ^m  om ) ( A. c  e.  om  ( b `  suc  c )  C_  (
b `  c )  ->  |^| ran  b  e. 
ran  b ) )
261fin23lem13 8513 . . . 4  |-  ( c  e.  om  ->  ( U `  suc  c ) 
C_  ( U `  c ) )
2726rgen 2793 . . 3  |-  A. c  e.  om  ( U `  suc  c )  C_  ( U `  c )
2827a1i 11 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  A. c  e.  om  ( U `  suc  c
)  C_  ( U `  c ) )
29 fveq1 5702 . . . . . 6  |-  ( b  =  U  ->  (
b `  suc  c )  =  ( U `  suc  c ) )
30 fveq1 5702 . . . . . 6  |-  ( b  =  U  ->  (
b `  c )  =  ( U `  c ) )
3129, 30sseq12d 3397 . . . . 5  |-  ( b  =  U  ->  (
( b `  suc  c )  C_  (
b `  c )  <->  ( U `  suc  c
)  C_  ( U `  c ) ) )
3231ralbidv 2747 . . . 4  |-  ( b  =  U  ->  ( A. c  e.  om  ( b `  suc  c )  C_  (
b `  c )  <->  A. c  e.  om  ( U `  suc  c ) 
C_  ( U `  c ) ) )
33 rneq 5077 . . . . . 6  |-  ( b  =  U  ->  ran  b  =  ran  U )
3433inteqd 4145 . . . . 5  |-  ( b  =  U  ->  |^| ran  b  =  |^| ran  U
)
3534, 33eleq12d 2511 . . . 4  |-  ( b  =  U  ->  ( |^| ran  b  e.  ran  b 
<-> 
|^| ran  U  e.  ran  U ) )
3632, 35imbi12d 320 . . 3  |-  ( b  =  U  ->  (
( A. c  e. 
om  ( b `  suc  c )  C_  (
b `  c )  ->  |^| ran  b  e. 
ran  b )  <->  ( A. c  e.  om  ( U `  suc  c ) 
C_  ( U `  c )  ->  |^| ran  U  e.  ran  U ) ) )
3736rspcv 3081 . 2  |-  ( U  e.  ( ~P U. ran  t  ^m  om )  ->  ( A. b  e.  ( ~P U. ran  t  ^m  om ) ( A. c  e.  om  ( b `  suc  c )  C_  (
b `  c )  ->  |^| ran  b  e. 
ran  b )  -> 
( A. c  e. 
om  ( U `  suc  c )  C_  ( U `  c )  ->  |^| ran  U  e. 
ran  U ) ) )
3821, 25, 28, 37syl3c 61 1  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2727   _Vcvv 2984    i^i cin 3339    C_ wss 3340   (/)c0 3649   ifcif 3803   ~Pcpw 3872   U.cuni 4103   |^|cint 4140   suc csuc 4733   ran crn 4853    Fn wfn 5425   -->wf 5426   -1-1->wf1 5427   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   omcom 6488  seq𝜔cseqom 6914    ^m cmap 7226
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-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-2nd 6590  df-recs 6844  df-rdg 6878  df-seqom 6915  df-map 7228
This theorem is referenced by:  fin23lem21  8520
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