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Theorem fin23lem17 7964
Description: Lemma for fin23 8015. 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 6467 . . . . 5  |-  U  Fn  om
3 dffn3 5396 . . . . 5  |-  ( U  Fn  om  <->  U : om
--> ran  U )
42, 3mpbi 199 . . . 4  |-  U : om
--> ran  U
5 pwuni 4206 . . . . 5  |-  ran  U  C_ 
~P U. ran  U
61fin23lem16 7961 . . . . . 6  |-  U. ran  U  =  U. ran  t
76pweqi 3629 . . . . 5  |-  ~P U. ran  U  =  ~P U. ran  t
85, 7sseqtri 3210 . . . 4  |-  ran  U  C_ 
~P U. ran  t
9 fss 5397 . . . 4  |-  ( ( U : om --> ran  U  /\  ran  U  C_  ~P U.
ran  t )  ->  U : om --> ~P U. ran  t )
104, 8, 9mp2an 653 . . 3  |-  U : om
--> ~P U. ran  t
11 vex 2791 . . . . . . 7  |-  t  e. 
_V
1211rnex 4942 . . . . . 6  |-  ran  t  e.  _V
1312uniex 4516 . . . . 5  |-  U. ran  t  e.  _V
1413pwex 4193 . . . 4  |-  ~P U. ran  t  e.  _V
15 f1f 5437 . . . . . 6  |-  ( t : om -1-1-> V  -> 
t : om --> V )
16 dmfex 5424 . . . . . 6  |-  ( ( t  e.  _V  /\  t : om --> V )  ->  om  e.  _V )
1711, 15, 16sylancr 644 . . . . 5  |-  ( t : om -1-1-> V  ->  om  e.  _V )
1817adantl 452 . . . 4  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  om  e.  _V )
19 elmapg 6785 . . . 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 644 . . 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 224 . 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 7955 . . . 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 232 . . 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 451 . 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 7958 . . . 4  |-  ( c  e.  om  ->  ( U `  suc  c ) 
C_  ( U `  c ) )
2726rgen 2608 . . 3  |-  A. c  e.  om  ( U `  suc  c )  C_  ( U `  c )
2827a1i 10 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  A. c  e.  om  ( U `  suc  c
)  C_  ( U `  c ) )
29 fveq1 5524 . . . . . 6  |-  ( b  =  U  ->  (
b `  suc  c )  =  ( U `  suc  c ) )
30 fveq1 5524 . . . . . 6  |-  ( b  =  U  ->  (
b `  c )  =  ( U `  c ) )
3129, 30sseq12d 3207 . . . . 5  |-  ( b  =  U  ->  (
( b `  suc  c )  C_  (
b `  c )  <->  ( U `  suc  c
)  C_  ( U `  c ) ) )
3231ralbidv 2563 . . . 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 4904 . . . . . 6  |-  ( b  =  U  ->  ran  b  =  ran  U )
3433inteqd 3867 . . . . 5  |-  ( b  =  U  ->  |^| ran  b  =  |^| ran  U
)
3534, 33eleq12d 2351 . . . 4  |-  ( b  =  U  ->  ( |^| ran  b  e.  ran  b 
<-> 
|^| ran  U  e.  ran  U ) )
3632, 35imbi12d 311 . . 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 2880 . 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 57 1  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   suc csuc 4394   omcom 4656   ran crn 4690    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459    ^m cmap 6772
This theorem is referenced by:  fin23lem21  7965
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-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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460  df-map 6774
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