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Theorem fin23lem21 8618
Description: Lemma for fin23 8668. 
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 8617 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
41fnseqom 7019 . . . . 5  |-  U  Fn  om
5 fvelrnb 5847 . . . . 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 3079 . . . . . . . . . 10  |-  t  e. 
_V
9 f1f1orn 5759 . . . . . . . . . 10  |-  ( t : om -1-1-> V  -> 
t : om -1-1-onto-> ran  t )
10 f1oen3g 7434 . . . . . . . . . 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 7635 . . . . . . . . 9  |-  -.  om  e.  Fin
13 ssdif0 3844 . . . . . . . . . . 11  |-  ( ran  t  C_  { (/) }  <->  ( ran  t  \  { (/) } )  =  (/) )
14 snfi 7499 . . . . . . . . . . . . 13  |-  { (/) }  e.  Fin
15 ssfi 7643 . . . . . . . . . . . . 13  |-  ( ( { (/) }  e.  Fin  /\ 
ran  t  C_  { (/) } )  ->  ran  t  e. 
Fin )
1614, 15mpan 670 . . . . . . . . . . . 12  |-  ( ran  t  C_  { (/) }  ->  ran  t  e.  Fin )
17 enfi 7639 . . . . . . . . . . . 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 2663 . . . . . . . . 9  |-  ( om 
~~  ran  t  ->  ( -.  om  e.  Fin  ->  ( ran  t  \  { (/) } )  =/=  (/) ) )
2111, 12, 20mpisyl 18 . . . . . . . 8  |-  ( t : om -1-1-> V  -> 
( ran  t  \  { (/) } )  =/=  (/) )
22 n0 3753 . . . . . . . . 9  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  <->  E. a 
a  e.  ( ran  t  \  { (/) } ) )
23 eldifsn 4107 . . . . . . . . . . 11  |-  ( a  e.  ( ran  t  \  { (/) } )  <->  ( a  e.  ran  t  /\  a  =/=  (/) ) )
24 elssuni 4228 . . . . . . . . . . . 12  |-  ( a  e.  ran  t  -> 
a  C_  U. ran  t
)
25 ssn0 3777 . . . . . . . . . . . 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 1689 . . . . . . . . 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 8612 . . . . . . 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 2732 . . . . . 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 2945 . . . 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 1370   E.wex 1587    e. wcel 1758   {cab 2439    =/= wne 2647   A.wral 2798   E.wrex 2799   _Vcvv 3076    \ cdif 3432    i^i cin 3434    C_ wss 3435   (/)c0 3744   ifcif 3898   ~Pcpw 3967   {csn 3984   U.cuni 4198   |^|cint 4235   class class class wbr 4399   suc csuc 4828   ran crn 4948    Fn wfn 5520   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   omcom 6585  seq𝜔cseqom 7011    ^m cmap 7323    ~~ cen 7416   Fincfn 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-2nd 6687  df-recs 6941  df-rdg 6975  df-seqom 7012  df-1o 7029  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423
This theorem is referenced by:  fin23lem31  8622
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