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Theorem fin23lem14 8765
Description: Lemma for fin23 8821. 
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 5879 . . . . 5  |-  ( a  =  (/)  ->  ( U `
 a )  =  ( U `  (/) ) )
21neeq1d 2702 . . . 4  |-  ( a  =  (/)  ->  ( ( U `  a )  =/=  (/)  <->  ( U `  (/) )  =/=  (/) ) )
32imbi2d 318 . . 3  |-  ( a  =  (/)  ->  ( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) ) ) )
4 fveq2 5879 . . . . 5  |-  ( a  =  b  ->  ( U `  a )  =  ( U `  b ) )
54neeq1d 2702 . . . 4  |-  ( a  =  b  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  b )  =/=  (/) ) )
65imbi2d 318 . . 3  |-  ( a  =  b  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) ) ) )
7 fveq2 5879 . . . . 5  |-  ( a  =  suc  b  -> 
( U `  a
)  =  ( U `
 suc  b )
)
87neeq1d 2702 . . . 4  |-  ( a  =  suc  b  -> 
( ( U `  a )  =/=  (/)  <->  ( U `  suc  b )  =/=  (/) ) )
98imbi2d 318 . . 3  |-  ( a  =  suc  b  -> 
( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
10 fveq2 5879 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
1110neeq1d 2702 . . . 4  |-  ( a  =  A  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  A )  =/=  (/) ) )
1211imbi2d 318 . . 3  |-  ( a  =  A  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) ) )
13 vex 3085 . . . . . . 7  |-  t  e. 
_V
1413rnex 6739 . . . . . 6  |-  ran  t  e.  _V
1514uniex 6599 . . . . 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 7179 . . . . 5  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1815, 17mp1i 13 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  = 
U. ran  t )
19 id 23 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  U. ran  t  =/=  (/) )
2018, 19eqnetrd 2718 . . 3  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) )
2116fin23lem12 8763 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  suc  b )  =  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) ) )
2221adantr 467 . . . . . 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 3916 . . . . . . . . 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 467 . . . . . . . 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 765 . . . . . . . 8  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  -> 
( U `  b
)  =/=  (/) )
2624, 25eqnetrd 2718 . . . . . . 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 3919 . . . . . . . . 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 467 . . . . . . . 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 2621 . . . . . . . . . 10  |-  ( ( ( t `  b
)  i^i  ( U `  b ) )  =/=  (/) 
<->  -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) )
3029biimpri 210 . . . . . . . . 9  |-  ( -.  ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  ->  (
( t `  b
)  i^i  ( U `  b ) )  =/=  (/) )
3130adantr 467 . . . . . . . 8  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  ( ( t `
 b )  i^i  ( U `  b
) )  =/=  (/) )
3228, 31eqnetrd 2718 . . . . . . 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 798 . . . . . 6  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =/=  (/) )
3422, 33eqnetrd 2718 . . . . 5  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  ( U `  suc  b )  =/=  (/) )
3534ex 436 . . . 4  |-  ( b  e.  om  ->  (
( U `  b
)  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) )
3635imim2d 55 . . 3  |-  ( b  e.  om  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) )  -> 
( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
373, 6, 9, 12, 20, 36finds 6731 . 2  |-  ( A  e.  om  ->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) )
3837imp 431 1  |-  ( ( A  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  A
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   _Vcvv 3082    i^i cin 3436   (/)c0 3762   ifcif 3910   U.cuni 4217   ran crn 4852   suc csuc 5442   ` cfv 5599    |-> cmpt2 6305   omcom 6704  seq𝜔cseqom 7170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-seqom 7171
This theorem is referenced by:  fin23lem21  8771
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