MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem21 Structured version   Unicode version

Theorem fin23lem21 8500
Description: Lemma for fin23 8550. 
X is not empty. We only need here that  t has at least one set in its range besides  (/); the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-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
fin23lem21  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| 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 fin23lem21
StepHypRef Expression
1 fin23lem.a . . 3  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
2 fin23lem17.f . . 3  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
31, 2fin23lem17 8499 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
41fnseqom 6902 . . . . 5  |-  U  Fn  om
5 fvelrnb 5734 . . . . 5  |-  ( U  Fn  om  ->  ( |^| ran  U  e.  ran  U  <->  E. a  e.  om  ( U `  a )  =  |^| ran  U
) )
64, 5ax-mp 5 . . . 4  |-  ( |^| ran 
U  e.  ran  U  <->  E. a  e.  om  ( U `  a )  =  |^| ran  U )
7 id 22 . . . . . . 7  |-  ( a  e.  om  ->  a  e.  om )
8 vex 2970 . . . . . . . . . 10  |-  t  e. 
_V
9 f1f1orn 5647 . . . . . . . . . 10  |-  ( t : om -1-1-> V  -> 
t : om -1-1-onto-> ran  t )
10 f1oen3g 7317 . . . . . . . . . 10  |-  ( ( t  e.  _V  /\  t : om -1-1-onto-> ran  t )  ->  om  ~~  ran  t )
118, 9, 10sylancr 663 . . . . . . . . 9  |-  ( t : om -1-1-> V  ->  om  ~~  ran  t )
12 ominf 7517 . . . . . . . . 9  |-  -.  om  e.  Fin
13 ssdif0 3732 . . . . . . . . . . 11  |-  ( ran  t  C_  { (/) }  <->  ( ran  t  \  { (/) } )  =  (/) )
14 snfi 7382 . . . . . . . . . . . . 13  |-  { (/) }  e.  Fin
15 ssfi 7525 . . . . . . . . . . . . 13  |-  ( ( { (/) }  e.  Fin  /\ 
ran  t  C_  { (/) } )  ->  ran  t  e. 
Fin )
1614, 15mpan 670 . . . . . . . . . . . 12  |-  ( ran  t  C_  { (/) }  ->  ran  t  e.  Fin )
17 enfi 7521 . . . . . . . . . . . 12  |-  ( om 
~~  ran  t  ->  ( om  e.  Fin  <->  ran  t  e. 
Fin ) )
1816, 17syl5ibr 221 . . . . . . . . . . 11  |-  ( om 
~~  ran  t  ->  ( ran  t  C_  { (/) }  ->  om  e.  Fin ) )
1913, 18syl5bir 218 . . . . . . . . . 10  |-  ( om 
~~  ran  t  ->  ( ( ran  t  \  { (/) } )  =  (/)  ->  om  e.  Fin ) )
2019necon3bd 2640 . . . . . . . . 9  |-  ( om 
~~  ran  t  ->  ( -.  om  e.  Fin  ->  ( ran  t  \  { (/) } )  =/=  (/) ) )
2111, 12, 20mpisyl 18 . . . . . . . 8  |-  ( t : om -1-1-> V  -> 
( ran  t  \  { (/) } )  =/=  (/) )
22 n0 3641 . . . . . . . . 9  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  <->  E. a 
a  e.  ( ran  t  \  { (/) } ) )
23 eldifsn 3995 . . . . . . . . . . 11  |-  ( a  e.  ( ran  t  \  { (/) } )  <->  ( a  e.  ran  t  /\  a  =/=  (/) ) )
24 elssuni 4116 . . . . . . . . . . . 12  |-  ( a  e.  ran  t  -> 
a  C_  U. ran  t
)
25 ssn0 3665 . . . . . . . . . . . 12  |-  ( ( a  C_  U. ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2624, 25sylan 471 . . . . . . . . . . 11  |-  ( ( a  e.  ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2723, 26sylbi 195 . . . . . . . . . 10  |-  ( a  e.  ( ran  t  \  { (/) } )  ->  U. ran  t  =/=  (/) )
2827exlimiv 1688 . . . . . . . . 9  |-  ( E. a  a  e.  ( ran  t  \  { (/)
} )  ->  U. ran  t  =/=  (/) )
2922, 28sylbi 195 . . . . . . . 8  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  ->  U. ran  t  =/=  (/) )
3021, 29syl 16 . . . . . . 7  |-  ( t : om -1-1-> V  ->  U. ran  t  =/=  (/) )
311fin23lem14 8494 . . . . . . 7  |-  ( ( a  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  a
)  =/=  (/) )
327, 30, 31syl2anr 478 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( U `  a )  =/=  (/) )
33 neeq1 2611 . . . . . 6  |-  ( ( U `  a )  =  |^| ran  U  ->  ( ( U `  a )  =/=  (/)  <->  |^| ran  U  =/=  (/) ) )
3432, 33syl5ibcom 220 . . . . 5  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( ( U `
 a )  = 
|^| ran  U  ->  |^|
ran  U  =/=  (/) ) )
3534rexlimdva 2836 . . . 4  |-  ( t : om -1-1-> V  -> 
( E. a  e. 
om  ( U `  a )  =  |^| ran 
U  ->  |^| ran  U  =/=  (/) ) )
366, 35syl5bi 217 . . 3  |-  ( t : om -1-1-> V  -> 
( |^| ran  U  e. 
ran  U  ->  |^| ran  U  =/=  (/) ) )
3736adantl 466 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  -> 
( |^| ran  U  e. 
ran  U  ->  |^| ran  U  =/=  (/) ) )
383, 37mpd 15 1  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424    =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967    \ cdif 3320    i^i cin 3322    C_ wss 3323   (/)c0 3632   ifcif 3786   ~Pcpw 3855   {csn 3872   U.cuni 4086   |^|cint 4123   class class class wbr 4287   suc csuc 4716   ran crn 4836    Fn wfn 5408   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   omcom 6471  seq𝜔cseqom 6894    ^m cmap 7206    ~~ cen 7299   Fincfn 7302
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-seqom 6895  df-1o 6912  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306
This theorem is referenced by:  fin23lem31  8504
  Copyright terms: Public domain W3C validator