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Theorem fin23lem14 7959
Description: Lemma for fin23 8015. 
U will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
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 )
Assertion
Ref Expression
fin23lem14  |-  ( ( A  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  A
)  =/=  (/) )
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    U( t)

Proof of Theorem fin23lem14
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( a  =  (/)  ->  ( U `
 a )  =  ( U `  (/) ) )
21neeq1d 2459 . . . 4  |-  ( a  =  (/)  ->  ( ( U `  a )  =/=  (/)  <->  ( U `  (/) )  =/=  (/) ) )
32imbi2d 307 . . 3  |-  ( a  =  (/)  ->  ( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) ) ) )
4 fveq2 5525 . . . . 5  |-  ( a  =  b  ->  ( U `  a )  =  ( U `  b ) )
54neeq1d 2459 . . . 4  |-  ( a  =  b  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  b )  =/=  (/) ) )
65imbi2d 307 . . 3  |-  ( a  =  b  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) ) ) )
7 fveq2 5525 . . . . 5  |-  ( a  =  suc  b  -> 
( U `  a
)  =  ( U `
 suc  b )
)
87neeq1d 2459 . . . 4  |-  ( a  =  suc  b  -> 
( ( U `  a )  =/=  (/)  <->  ( U `  suc  b )  =/=  (/) ) )
98imbi2d 307 . . 3  |-  ( a  =  suc  b  -> 
( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
10 fveq2 5525 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
1110neeq1d 2459 . . . 4  |-  ( a  =  A  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  A )  =/=  (/) ) )
1211imbi2d 307 . . 3  |-  ( a  =  A  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) ) )
13 vex 2791 . . . . . . 7  |-  t  e. 
_V
1413rnex 4942 . . . . . 6  |-  ran  t  e.  _V
1514uniex 4516 . . . . 5  |-  U. ran  t  e.  _V
16 fin23lem.a . . . . . 6  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
1716seqom0g 6468 . . . . 5  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1815, 17mp1i 11 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  = 
U. ran  t )
19 id 19 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  U. ran  t  =/=  (/) )
2018, 19eqnetrd 2464 . . 3  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) )
2116fin23lem12 7957 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  suc  b )  =  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) ) )
2221adantr 451 . . . . . 6  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  ( U `  suc  b )  =  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) ) )
23 iftrue 3571 . . . . . . . . 9  |-  ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/)  ->  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) )  =  ( U `  b ) )
2423adantr 451 . . . . . . . 8  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =  ( U `  b ) )
25 simprr 733 . . . . . . . 8  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  -> 
( U `  b
)  =/=  (/) )
2624, 25eqnetrd 2464 . . . . . . 7  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =/=  (/) )
27 iffalse 3572 . . . . . . . . 9  |-  ( -.  ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =  ( ( t `
 b )  i^i  ( U `  b
) ) )
2827adantr 451 . . . . . . . 8  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) )  =  ( ( t `
 b )  i^i  ( U `  b
) ) )
29 df-ne 2448 . . . . . . . . . 10  |-  ( ( ( t `  b
)  i^i  ( U `  b ) )  =/=  (/) 
<->  -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) )
3029biimpri 197 . . . . . . . . 9  |-  ( -.  ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  ->  (
( t `  b
)  i^i  ( U `  b ) )  =/=  (/) )
3130adantr 451 . . . . . . . 8  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  ( ( t `
 b )  i^i  ( U `  b
) )  =/=  (/) )
3228, 31eqnetrd 2464 . . . . . . 7  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) )  =/=  (/) )
3326, 32pm2.61ian 765 . . . . . 6  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =/=  (/) )
3422, 33eqnetrd 2464 . . . . 5  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  ( U `  suc  b )  =/=  (/) )
3534ex 423 . . . 4  |-  ( b  e.  om  ->  (
( U `  b
)  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) )
3635imim2d 48 . . 3  |-  ( b  e.  om  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) )  -> 
( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
373, 6, 9, 12, 20, 36finds 4682 . 2  |-  ( A  e.  om  ->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) )
3837imp 418 1  |-  ( ( A  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  A
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ifcif 3565   U.cuni 3827   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255    e. cmpt2 5860  seq𝜔cseqom 6459
This theorem is referenced by:  fin23lem21  7965
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-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
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