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Theorem fin23lem21 7965
Description: Lemma for fin23 8015. 
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 7964 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
41fnseqom 6467 . . . . 5  |-  U  Fn  om
5 fvelrnb 5570 . . . . 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 19 . . . . . . 7  |-  ( a  e.  om  ->  a  e.  om )
8 vex 2791 . . . . . . . . . 10  |-  t  e. 
_V
9 f1f1orn 5483 . . . . . . . . . 10  |-  ( t : om -1-1-> V  -> 
t : om -1-1-onto-> ran  t )
10 f1oen3g 6877 . . . . . . . . . 10  |-  ( ( t  e.  _V  /\  t : om -1-1-onto-> ran  t )  ->  om  ~~  ran  t )
118, 9, 10sylancr 644 . . . . . . . . 9  |-  ( t : om -1-1-> V  ->  om  ~~  ran  t )
12 ominf 7075 . . . . . . . . 9  |-  -.  om  e.  Fin
13 ssdif0 3513 . . . . . . . . . . 11  |-  ( ran  t  C_  { (/) }  <->  ( ran  t  \  { (/) } )  =  (/) )
14 snfi 6941 . . . . . . . . . . . . 13  |-  { (/) }  e.  Fin
15 ssfi 7083 . . . . . . . . . . . . 13  |-  ( ( { (/) }  e.  Fin  /\ 
ran  t  C_  { (/) } )  ->  ran  t  e. 
Fin )
1614, 15mpan 651 . . . . . . . . . . . 12  |-  ( ran  t  C_  { (/) }  ->  ran  t  e.  Fin )
17 enfi 7079 . . . . . . . . . . . 12  |-  ( om 
~~  ran  t  ->  ( om  e.  Fin  <->  ran  t  e. 
Fin ) )
1816, 17syl5ibr 212 . . . . . . . . . . 11  |-  ( om 
~~  ran  t  ->  ( ran  t  C_  { (/) }  ->  om  e.  Fin ) )
1913, 18syl5bir 209 . . . . . . . . . 10  |-  ( om 
~~  ran  t  ->  ( ( ran  t  \  { (/) } )  =  (/)  ->  om  e.  Fin ) )
2019necon3bd 2483 . . . . . . . . 9  |-  ( om 
~~  ran  t  ->  ( -.  om  e.  Fin  ->  ( ran  t  \  { (/) } )  =/=  (/) ) )
2111, 12, 20ee10 1366 . . . . . . . 8  |-  ( t : om -1-1-> V  -> 
( ran  t  \  { (/) } )  =/=  (/) )
22 n0 3464 . . . . . . . . 9  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  <->  E. a 
a  e.  ( ran  t  \  { (/) } ) )
23 eldifsn 3749 . . . . . . . . . . 11  |-  ( a  e.  ( ran  t  \  { (/) } )  <->  ( a  e.  ran  t  /\  a  =/=  (/) ) )
24 elssuni 3855 . . . . . . . . . . . 12  |-  ( a  e.  ran  t  -> 
a  C_  U. ran  t
)
25 ssn0 3487 . . . . . . . . . . . 12  |-  ( ( a  C_  U. ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2624, 25sylan 457 . . . . . . . . . . 11  |-  ( ( a  e.  ran  t  /\  a  =/=  (/) )  ->  U. ran  t  =/=  (/) )
2723, 26sylbi 187 . . . . . . . . . 10  |-  ( a  e.  ( ran  t  \  { (/) } )  ->  U. ran  t  =/=  (/) )
2827exlimiv 1666 . . . . . . . . 9  |-  ( E. a  a  e.  ( ran  t  \  { (/)
} )  ->  U. ran  t  =/=  (/) )
2922, 28sylbi 187 . . . . . . . 8  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  ->  U. ran  t  =/=  (/) )
3021, 29syl 15 . . . . . . 7  |-  ( t : om -1-1-> V  ->  U. ran  t  =/=  (/) )
311fin23lem14 7959 . . . . . . 7  |-  ( ( a  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  a
)  =/=  (/) )
327, 30, 31syl2anr 464 . . . . . 6  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( U `  a )  =/=  (/) )
33 neeq1 2454 . . . . . 6  |-  ( ( U `  a )  =  |^| ran  U  ->  ( ( U `  a )  =/=  (/)  <->  |^| ran  U  =/=  (/) ) )
3432, 33syl5ibcom 211 . . . . 5  |-  ( ( t : om -1-1-> V  /\  a  e.  om )  ->  ( ( U `
 a )  = 
|^| ran  U  ->  |^|
ran  U  =/=  (/) ) )
3534rexlimdva 2667 . . . 4  |-  ( t : om -1-1-> V  -> 
( E. a  e. 
om  ( U `  a )  =  |^| ran 
U  ->  |^| ran  U  =/=  (/) ) )
366, 35syl5bi 208 . . 3  |-  ( t : om -1-1-> V  -> 
( |^| ran  U  e. 
ran  U  ->  |^| ran  U  =/=  (/) ) )
3736adantl 452 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  -> 
( |^| ran  U  e. 
ran  U  ->  |^| ran  U  =/=  (/) ) )
383, 37mpd 14 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640   U.cuni 3827   |^|cint 3862   class class class wbr 4023   suc csuc 4394   omcom 4656   ran crn 4690    Fn wfn 5250   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459    ^m cmap 6772    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  fin23lem31  7969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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