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Theorem fin23lem14 8626
Description: Lemma for fin23 8682. 
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 5774 . . . . 5  |-  ( a  =  (/)  ->  ( U `
 a )  =  ( U `  (/) ) )
21neeq1d 2659 . . . 4  |-  ( a  =  (/)  ->  ( ( U `  a )  =/=  (/)  <->  ( U `  (/) )  =/=  (/) ) )
32imbi2d 314 . . 3  |-  ( a  =  (/)  ->  ( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) ) ) )
4 fveq2 5774 . . . . 5  |-  ( a  =  b  ->  ( U `  a )  =  ( U `  b ) )
54neeq1d 2659 . . . 4  |-  ( a  =  b  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  b )  =/=  (/) ) )
65imbi2d 314 . . 3  |-  ( a  =  b  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) ) ) )
7 fveq2 5774 . . . . 5  |-  ( a  =  suc  b  -> 
( U `  a
)  =  ( U `
 suc  b )
)
87neeq1d 2659 . . . 4  |-  ( a  =  suc  b  -> 
( ( U `  a )  =/=  (/)  <->  ( U `  suc  b )  =/=  (/) ) )
98imbi2d 314 . . 3  |-  ( a  =  suc  b  -> 
( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  suc  b )  =/=  (/) ) ) )
10 fveq2 5774 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
1110neeq1d 2659 . . . 4  |-  ( a  =  A  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  A )  =/=  (/) ) )
1211imbi2d 314 . . 3  |-  ( a  =  A  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) ) )
13 vex 3037 . . . . . . 7  |-  t  e. 
_V
1413rnex 6633 . . . . . 6  |-  ran  t  e.  _V
1514uniex 6495 . . . . 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 7039 . . . . 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 2675 . . 3  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) )
2116fin23lem12 8624 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  suc  b )  =  if ( ( ( t `  b
)  i^i  ( U `  b ) )  =  (/) ,  ( U `  b ) ,  ( ( t `  b
)  i^i  ( U `  b ) ) ) )
2221adantr 463 . . . . . 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 3863 . . . . . . . . 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 463 . . . . . . . 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 755 . . . . . . . 8  |-  ( ( ( ( t `  b )  i^i  ( U `  b )
)  =  (/)  /\  (
b  e.  om  /\  ( U `  b )  =/=  (/) ) )  -> 
( U `  b
)  =/=  (/) )
2624, 25eqnetrd 2675 . . . . . . 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 3866 . . . . . . . . 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 463 . . . . . . . 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 2579 . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( -.  ( ( t `
 b )  i^i  ( U `  b
) )  =  (/)  /\  ( b  e.  om  /\  ( U `  b
)  =/=  (/) ) )  ->  ( ( t `
 b )  i^i  ( U `  b
) )  =/=  (/) )
3228, 31eqnetrd 2675 . . . . . . 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 2675 . . . . 5  |-  ( ( b  e.  om  /\  ( U `  b )  =/=  (/) )  ->  ( U `  suc  b )  =/=  (/) )
3534ex 432 . . . 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 6625 . 2  |-  ( A  e.  om  ->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) )
3837imp 427 1  |-  ( ( A  e.  om  /\  U.
ran  t  =/=  (/) )  -> 
( U `  A
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   _Vcvv 3034    i^i cin 3388   (/)c0 3711   ifcif 3857   U.cuni 4163   suc csuc 4794   ran crn 4914   ` cfv 5496    |-> cmpt2 6198   omcom 6599  seq𝜔cseqom 7030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-seqom 7031
This theorem is referenced by:  fin23lem21  8632
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