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Theorem fin23lem17 8719
Description: Lemma for fin23 8770. 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 7121 . . . . 5  |-  U  Fn  om
3 dffn3 5738 . . . . 5  |-  ( U  Fn  om  <->  U : om
--> ran  U )
42, 3mpbi 208 . . . 4  |-  U : om
--> ran  U
5 pwuni 4678 . . . . 5  |-  ran  U  C_ 
~P U. ran  U
61fin23lem16 8716 . . . . . 6  |-  U. ran  U  =  U. ran  t
76pweqi 4014 . . . . 5  |-  ~P U. ran  U  =  ~P U. ran  t
85, 7sseqtri 3536 . . . 4  |-  ran  U  C_ 
~P U. ran  t
9 fss 5739 . . . 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 3116 . . . . . . 7  |-  t  e. 
_V
1211rnex 6719 . . . . . 6  |-  ran  t  e.  _V
1312uniex 6581 . . . . 5  |-  U. ran  t  e.  _V
1413pwex 4630 . . . 4  |-  ~P U. ran  t  e.  _V
15 f1f 5781 . . . . . 6  |-  ( t : om -1-1-> V  -> 
t : om --> V )
16 dmfex 6743 . . . . . 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 7434 . . . 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 8710 . . . 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 8713 . . . 4  |-  ( c  e.  om  ->  ( U `  suc  c ) 
C_  ( U `  c ) )
2726rgen 2824 . . 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 5865 . . . . . 6  |-  ( b  =  U  ->  (
b `  suc  c )  =  ( U `  suc  c ) )
30 fveq1 5865 . . . . . 6  |-  ( b  =  U  ->  (
b `  c )  =  ( U `  c ) )
3129, 30sseq12d 3533 . . . . 5  |-  ( b  =  U  ->  (
( b `  suc  c )  C_  (
b `  c )  <->  ( U `  suc  c
)  C_  ( U `  c ) ) )
3231ralbidv 2903 . . . 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 5228 . . . . . 6  |-  ( b  =  U  ->  ran  b  =  ran  U )
3433inteqd 4287 . . . . 5  |-  ( b  =  U  ->  |^| ran  b  =  |^| ran  U
)
3534, 33eleq12d 2549 . . . 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 3210 . 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   suc csuc 4880   ran crn 5000    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   omcom 6685  seq𝜔cseqom 7113    ^m cmap 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-seqom 7114  df-map 7423
This theorem is referenced by:  fin23lem21  8720
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