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Theorem fin23lem21 8211
Description: Lemma for fin23 8261. 
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 8210 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
41fnseqom 6704 . . . . 5  |-  U  Fn  om
5 fvelrnb 5766 . . . . 5  |-  ( U  Fn  om  ->  ( |^| ran  U  e.  ran  U  <->  E. a  e.  om  ( U `  a )  =  |^| ran  U
) )
64, 5ax-mp 8 . . . 4  |-  ( |^| ran 
U  e.  ran  U  <->  E. a  e.  om  ( U `  a )  =  |^| ran  U )
7 id 20 . . . . . . 7  |-  ( a  e.  om  ->  a  e.  om )
8 vex 2951 . . . . . . . . . 10  |-  t  e. 
_V
9 f1f1orn 5677 . . . . . . . . . 10  |-  ( t : om -1-1-> V  -> 
t : om -1-1-onto-> ran  t )
10 f1oen3g 7115 . . . . . . . . . 10  |-  ( ( t  e.  _V  /\  t : om -1-1-onto-> ran  t )  ->  om  ~~  ran  t )
118, 9, 10sylancr 645 . . . . . . . . 9  |-  ( t : om -1-1-> V  ->  om  ~~  ran  t )
12 ominf 7313 . . . . . . . . 9  |-  -.  om  e.  Fin
13 ssdif0 3678 . . . . . . . . . . 11  |-  ( ran  t  C_  { (/) }  <->  ( ran  t  \  { (/) } )  =  (/) )
14 snfi 7179 . . . . . . . . . . . . 13  |-  { (/) }  e.  Fin
15 ssfi 7321 . . . . . . . . . . . . 13  |-  ( ( { (/) }  e.  Fin  /\ 
ran  t  C_  { (/) } )  ->  ran  t  e. 
Fin )
1614, 15mpan 652 . . . . . . . . . . . 12  |-  ( ran  t  C_  { (/) }  ->  ran  t  e.  Fin )
17 enfi 7317 . . . . . . . . . . . 12  |-  ( om 
~~  ran  t  ->  ( om  e.  Fin  <->  ran  t  e. 
Fin ) )
1816, 17syl5ibr 213 . . . . . . . . . . 11  |-  ( om 
~~  ran  t  ->  ( ran  t  C_  { (/) }  ->  om  e.  Fin ) )
1913, 18syl5bir 210 . . . . . . . . . 10  |-  ( om 
~~  ran  t  ->  ( ( ran  t  \  { (/) } )  =  (/)  ->  om  e.  Fin ) )
2019necon3bd 2635 . . . . . . . . 9  |-  ( om 
~~  ran  t  ->  ( -.  om  e.  Fin  ->  ( ran  t  \  { (/) } )  =/=  (/) ) )
2111, 12, 20ee10 1385 . . . . . . . 8  |-  ( t : om -1-1-> V  -> 
( ran  t  \  { (/) } )  =/=  (/) )
22 n0 3629 . . . . . . . . 9  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  <->  E. a 
a  e.  ( ran  t  \  { (/) } ) )
23 eldifsn 3919 . . . . . . . . . . 11  |-  ( a  e.  ( ran  t  \  { (/) } )  <->  ( a  e.  ran  t  /\  a  =/=  (/) ) )
24 elssuni 4035 . . . . . . . . . . . 12  |-  ( a  e.  ran  t  -> 
a  C_  U. ran  t
)
25 ssn0 3652 . . . . . . . . . . . 12  |-  ( ( a  C_  U. ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2624, 25sylan 458 . . . . . . . . . . 11  |-  ( ( a  e.  ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2723, 26sylbi 188 . . . . . . . . . 10  |-  ( a  e.  ( ran  t  \  { (/) } )  ->  U. ran  t  =/=  (/) )
2827exlimiv 1644 . . . . . . . . 9  |-  ( E. a  a  e.  ( ran  t  \  { (/)
} )  ->  U. ran  t  =/=  (/) )
2922, 28sylbi 188 . . . . . . . 8  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  ->  U. ran  t  =/=  (/) )
3021, 29syl 16 . . . . . . 7  |-  ( t : om -1-1-> V  ->  U. ran  t  =/=  (/) )
311fin23lem14 8205 . . . . . . 7  |-  ( ( a  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  a
)  =/=  (/) )
327, 30, 31syl2anr 465 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( U `  a )  =/=  (/) )
33 neeq1 2606 . . . . . 6  |-  ( ( U `  a )  =  |^| ran  U  ->  ( ( U `  a )  =/=  (/)  <->  |^| ran  U  =/=  (/) ) )
3432, 33syl5ibcom 212 . . . . 5  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( ( U `
 a )  = 
|^| ran  U  ->  |^|
ran  U  =/=  (/) ) )
3534rexlimdva 2822 . . . 4  |-  ( t : om -1-1-> V  -> 
( E. a  e. 
om  ( U `  a )  =  |^| ran 
U  ->  |^| ran  U  =/=  (/) ) )
366, 35syl5bi 209 . . 3  |-  ( t : om -1-1-> V  -> 
( |^| ran  U  e. 
ran  U  ->  |^| ran  U  =/=  (/) ) )
3736adantl 453 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   {csn 3806   U.cuni 4007   |^|cint 4042   class class class wbr 4204   suc csuc 4575   omcom 4837   ran crn 4871    Fn wfn 5441   -1-1->wf1 5443   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075  seq𝜔cseqom 6696    ^m cmap 7010    ~~ cen 7098   Fincfn 7101
This theorem is referenced by:  fin23lem31  8215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105
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