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Theorem fin23lem17 8009
Description: Lemma for fin23 8060. 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 6509 . . . . 5  |-  U  Fn  om
3 dffn3 5434 . . . . 5  |-  ( U  Fn  om  <->  U : om
--> ran  U )
42, 3mpbi 199 . . . 4  |-  U : om
--> ran  U
5 pwuni 4243 . . . . 5  |-  ran  U  C_ 
~P U. ran  U
61fin23lem16 8006 . . . . . 6  |-  U. ran  U  =  U. ran  t
76pweqi 3663 . . . . 5  |-  ~P U. ran  U  =  ~P U. ran  t
85, 7sseqtri 3244 . . . 4  |-  ran  U  C_ 
~P U. ran  t
9 fss 5435 . . . 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 2825 . . . . . . 7  |-  t  e. 
_V
1211rnex 4979 . . . . . 6  |-  ran  t  e.  _V
1312uniex 4553 . . . . 5  |-  U. ran  t  e.  _V
1413pwex 4230 . . . 4  |-  ~P U. ran  t  e.  _V
15 f1f 5475 . . . . . 6  |-  ( t : om -1-1-> V  -> 
t : om --> V )
16 dmfex 5462 . . . . . 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 6828 . . . 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 8000 . . . 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 8003 . . . 4  |-  ( c  e.  om  ->  ( U `  suc  c ) 
C_  ( U `  c ) )
2726rgen 2642 . . 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 5562 . . . . . 6  |-  ( b  =  U  ->  (
b `  suc  c )  =  ( U `  suc  c ) )
30 fveq1 5562 . . . . . 6  |-  ( b  =  U  ->  (
b `  c )  =  ( U `  c ) )
3129, 30sseq12d 3241 . . . . 5  |-  ( b  =  U  ->  (
( b `  suc  c )  C_  (
b `  c )  <->  ( U `  suc  c
)  C_  ( U `  c ) ) )
3231ralbidv 2597 . . . 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 4941 . . . . . 6  |-  ( b  =  U  ->  ran  b  =  ran  U )
3433inteqd 3904 . . . . 5  |-  ( b  =  U  ->  |^| ran  b  =  |^| ran  U
)
3534, 33eleq12d 2384 . . . 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 2914 . 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 1633    e. wcel 1701   {cab 2302   A.wral 2577   _Vcvv 2822    i^i cin 3185    C_ wss 3186   (/)c0 3489   ifcif 3599   ~Pcpw 3659   U.cuni 3864   |^|cint 3899   suc csuc 4431   omcom 4693   ran crn 4727    Fn wfn 5287   -->wf 5288   -1-1->wf1 5289   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902  seq𝜔cseqom 6501    ^m cmap 6815
This theorem is referenced by:  fin23lem21  8010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-recs 6430  df-rdg 6465  df-seqom 6502  df-map 6817
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