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Theorem fin23lem14 8709
Description: Lemma for fin23 8765. 
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 5864 . . . . 5  |-  ( a  =  (/)  ->  ( U `
 a )  =  ( U `  (/) ) )
21neeq1d 2744 . . . 4  |-  ( a  =  (/)  ->  ( ( U `  a )  =/=  (/)  <->  ( U `  (/) )  =/=  (/) ) )
32imbi2d 316 . . 3  |-  ( a  =  (/)  ->  ( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) ) ) )
4 fveq2 5864 . . . . 5  |-  ( a  =  b  ->  ( U `  a )  =  ( U `  b ) )
54neeq1d 2744 . . . 4  |-  ( a  =  b  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  b )  =/=  (/) ) )
65imbi2d 316 . . 3  |-  ( a  =  b  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) ) ) )
7 fveq2 5864 . . . . 5  |-  ( a  =  suc  b  -> 
( U `  a
)  =  ( U `
 suc  b )
)
87neeq1d 2744 . . . 4  |-  ( a  =  suc  b  -> 
( ( U `  a )  =/=  (/)  <->  ( U `  suc  b )  =/=  (/) ) )
98imbi2d 316 . . 3  |-  ( a  =  suc  b  -> 
( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
10 fveq2 5864 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
1110neeq1d 2744 . . . 4  |-  ( a  =  A  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  A )  =/=  (/) ) )
1211imbi2d 316 . . 3  |-  ( a  =  A  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) ) )
13 vex 3116 . . . . . . 7  |-  t  e. 
_V
1413rnex 6715 . . . . . 6  |-  ran  t  e.  _V
1514uniex 6578 . . . . 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 7118 . . . . 5  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1815, 17mp1i 12 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  = 
U. ran  t )
19 id 22 . . . 4  |-  ( U. ran  t  =/=  (/)  ->  U. ran  t  =/=  (/) )
2018, 19eqnetrd 2760 . . 3  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) )
2116fin23lem12 8707 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  suc  b )  =  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) ) )
2221adantr 465 . . . . . 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 3945 . . . . . . . . 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 465 . . . . . . . 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 756 . . . . . . . 8  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  -> 
( U `  b
)  =/=  (/) )
2624, 25eqnetrd 2760 . . . . . . 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 3948 . . . . . . . . 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 465 . . . . . . . 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 2664 . . . . . . . . . 10  |-  ( ( ( t `  b
)  i^i  ( U `  b ) )  =/=  (/) 
<->  -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) )
3029biimpri 206 . . . . . . . . 9  |-  ( -.  ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  ->  (
( t `  b
)  i^i  ( U `  b ) )  =/=  (/) )
3130adantr 465 . . . . . . . 8  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  ( ( t `
 b )  i^i  ( U `  b
) )  =/=  (/) )
3228, 31eqnetrd 2760 . . . . . . 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 788 . . . . . 6  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  if ( ( ( t `
 b )  i^i  ( U `  b
) )  =  (/) ,  ( U `  b
) ,  ( ( t `  b )  i^i  ( U `  b ) ) )  =/=  (/) )
3422, 33eqnetrd 2760 . . . . 5  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  ( U `  suc  b )  =/=  (/) )
3534ex 434 . . . 4  |-  ( b  e.  om  ->  (
( U `  b
)  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) )
3635imim2d 52 . . 3  |-  ( b  e.  om  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) )  -> 
( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
373, 6, 9, 12, 20, 36finds 6704 . 2  |-  ( A  e.  om  ->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) )
3837imp 429 1  |-  ( ( A  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  A
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    i^i cin 3475   (/)c0 3785   ifcif 3939   U.cuni 4245   suc csuc 4880   ran crn 5000   ` cfv 5586    |-> cmpt2 6284   omcom 6678  seq𝜔cseqom 7109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110
This theorem is referenced by:  fin23lem21  8715
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