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Theorem fin23lem21 7981
Description: Lemma for fin23 8031. 
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 7980 . 2  |-  ( ( U. ran  t  e.  F  /\  t : om -1-1-> V )  ->  |^| ran  U  e.  ran  U )
41fnseqom 6483 . . . . 5  |-  U  Fn  om
5 fvelrnb 5586 . . . . 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 2804 . . . . . . . . . 10  |-  t  e. 
_V
9 f1f1orn 5499 . . . . . . . . . 10  |-  ( t : om -1-1-> V  -> 
t : om -1-1-onto-> ran  t )
10 f1oen3g 6893 . . . . . . . . . 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 7091 . . . . . . . . 9  |-  -.  om  e.  Fin
13 ssdif0 3526 . . . . . . . . . . 11  |-  ( ran  t  C_  { (/) }  <->  ( ran  t  \  { (/) } )  =  (/) )
14 snfi 6957 . . . . . . . . . . . . 13  |-  { (/) }  e.  Fin
15 ssfi 7099 . . . . . . . . . . . . 13  |-  ( ( { (/) }  e.  Fin  /\ 
ran  t  C_  { (/) } )  ->  ran  t  e. 
Fin )
1614, 15mpan 651 . . . . . . . . . . . 12  |-  ( ran  t  C_  { (/) }  ->  ran  t  e.  Fin )
17 enfi 7095 . . . . . . . . . . . 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 2496 . . . . . . . . 9  |-  ( om 
~~  ran  t  ->  ( -.  om  e.  Fin  ->  ( ran  t  \  { (/) } )  =/=  (/) ) )
2111, 12, 20ee10 1366 . . . . . . . 8  |-  ( t : om -1-1-> V  -> 
( ran  t  \  { (/) } )  =/=  (/) )
22 n0 3477 . . . . . . . . 9  |-  ( ( ran  t  \  { (/)
} )  =/=  (/)  <->  E. a 
a  e.  ( ran  t  \  { (/) } ) )
23 eldifsn 3762 . . . . . . . . . . 11  |-  ( a  e.  ( ran  t  \  { (/) } )  <->  ( a  e.  ran  t  /\  a  =/=  (/) ) )
24 elssuni 3871 . . . . . . . . . . . 12  |-  ( a  e.  ran  t  -> 
a  C_  U. ran  t
)
25 ssn0 3500 . . . . . . . . . . . 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 1624 . . . . . . . . 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 7975 . . . . . . 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 2467 . . . . . 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 2680 . . . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   |^|cint 3878   class class class wbr 4039   suc csuc 4410   omcom 4672   ran crn 4706    Fn wfn 5266   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  seq𝜔cseqom 6475    ^m cmap 6788    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  fin23lem31  7985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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