MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem14 Unicode version

Theorem fin23lem14 7975
Description: Lemma for fin23 8031. 
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 5541 . . . . 5  |-  ( a  =  (/)  ->  ( U `
 a )  =  ( U `  (/) ) )
21neeq1d 2472 . . . 4  |-  ( a  =  (/)  ->  ( ( U `  a )  =/=  (/)  <->  ( U `  (/) )  =/=  (/) ) )
32imbi2d 307 . . 3  |-  ( a  =  (/)  ->  ( ( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) ) ) )
4 fveq2 5541 . . . . 5  |-  ( a  =  b  ->  ( U `  a )  =  ( U `  b ) )
54neeq1d 2472 . . . 4  |-  ( a  =  b  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  b )  =/=  (/) ) )
65imbi2d 307 . . 3  |-  ( a  =  b  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  b )  =/=  (/) ) ) )
7 fveq2 5541 . . . . 5  |-  ( a  =  suc  b  -> 
( U `  a
)  =  ( U `
 suc  b )
)
87neeq1d 2472 . . . 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 5541 . . . . 5  |-  ( a  =  A  ->  ( U `  a )  =  ( U `  A ) )
1110neeq1d 2472 . . . 4  |-  ( a  =  A  ->  (
( U `  a
)  =/=  (/)  <->  ( U `  A )  =/=  (/) ) )
1211imbi2d 307 . . 3  |-  ( a  =  A  ->  (
( U. ran  t  =/=  (/)  ->  ( U `  a )  =/=  (/) )  <->  ( U. ran  t  =/=  (/)  ->  ( U `  A )  =/=  (/) ) ) )
13 vex 2804 . . . . . . 7  |-  t  e. 
_V
1413rnex 4958 . . . . . 6  |-  ran  t  e.  _V
1514uniex 4532 . . . . 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 6484 . . . . 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 2477 . . 3  |-  ( U. ran  t  =/=  (/)  ->  ( U `  (/) )  =/=  (/) )
2116fin23lem12 7973 . . . . . . 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 3584 . . . . . . . . 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 2477 . . . . . . 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 3585 . . . . . . . . 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 2461 . . . . . . . . . 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 2477 . . . . . . 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 2477 . . . . 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 4698 . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    i^i cin 3164   (/)c0 3468   ifcif 3578   U.cuni 3843   suc csuc 4410   omcom 4672   ran crn 4706   ` cfv 5271    e. cmpt2 5876  seq𝜔cseqom 6475
This theorem is referenced by:  fin23lem21  7981
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-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
  Copyright terms: Public domain W3C validator