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Theorem fin23lem17 8735
Description: Lemma for fin23 8786. 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 7138 . . . . 5  |-  U  Fn  om
3 dffn3 5744 . . . . 5  |-  ( U  Fn  om  <->  U : om
--> ran  U )
42, 3mpbi 208 . . . 4  |-  U : om
--> ran  U
5 pwuni 4687 . . . . 5  |-  ran  U  C_ 
~P U. ran  U
61fin23lem16 8732 . . . . . 6  |-  U. ran  U  =  U. ran  t
76pweqi 4019 . . . . 5  |-  ~P U. ran  U  =  ~P U. ran  t
85, 7sseqtri 3531 . . . 4  |-  ran  U  C_ 
~P U. ran  t
9 fss 5745 . . . 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 3112 . . . . . . 7  |-  t  e. 
_V
1211rnex 6733 . . . . . 6  |-  ran  t  e.  _V
1312uniex 6595 . . . . 5  |-  U. ran  t  e.  _V
1413pwex 4639 . . . 4  |-  ~P U. ran  t  e.  _V
15 f1f 5787 . . . . . 6  |-  ( t : om -1-1-> V  -> 
t : om --> V )
16 dmfex 6757 . . . . . 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 7451 . . . 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 8726 . . . 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 8729 . . . 4  |-  ( c  e.  om  ->  ( U `  suc  c ) 
C_  ( U `  c ) )
2726rgen 2817 . . 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 5871 . . . . . 6  |-  ( b  =  U  ->  (
b `  suc  c )  =  ( U `  suc  c ) )
30 fveq1 5871 . . . . . 6  |-  ( b  =  U  ->  (
b `  c )  =  ( U `  c ) )
3129, 30sseq12d 3528 . . . . 5  |-  ( b  =  U  ->  (
( b `  suc  c )  C_  (
b `  c )  <->  ( U `  suc  c
)  C_  ( U `  c ) ) )
3231ralbidv 2896 . . . 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 5238 . . . . . 6  |-  ( b  =  U  ->  ran  b  =  ran  U )
3433inteqd 4293 . . . . 5  |-  ( b  =  U  ->  |^| ran  b  =  |^| ran  U
)
3534, 33eleq12d 2539 . . . 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 3206 . 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 1395    e. wcel 1819   {cab 2442   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   U.cuni 4251   |^|cint 4288   suc csuc 4889   ran crn 5009    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130    ^m cmap 7438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-seqom 7131  df-map 7440
This theorem is referenced by:  fin23lem21  8736
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