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Theorem fin23lem16 8220
Description: Lemma for fin23 8274. 
U ranges over the original set; in particular  ran  U is a set, although we do not assume here that  U is. (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
fin23lem16  |-  U. ran  U  =  U. ran  t
Distinct variable groups:    t, i, u    U, i, u
Allowed substitution hint:    U( t)

Proof of Theorem fin23lem16
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unissb 4047 . . 3  |-  ( U. ran  U  C_  U. ran  t  <->  A. a  e.  ran  U  a  C_  U. ran  t
)
2 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 )
32fnseqom 6715 . . . . 5  |-  U  Fn  om
4 fvelrnb 5777 . . . . 5  |-  ( U  Fn  om  ->  (
a  e.  ran  U  <->  E. b  e.  om  ( U `  b )  =  a ) )
53, 4ax-mp 5 . . . 4  |-  ( a  e.  ran  U  <->  E. b  e.  om  ( U `  b )  =  a )
6 peano1 4867 . . . . . . . 8  |-  (/)  e.  om
7 0ss 3658 . . . . . . . . 9  |-  (/)  C_  b
82fin23lem15 8219 . . . . . . . . 9  |-  ( ( ( b  e.  om  /\  (/)  e.  om )  /\  (/)  C_  b )  ->  ( U `  b )  C_  ( U `  (/) ) )
97, 8mpan2 654 . . . . . . . 8  |-  ( ( b  e.  om  /\  (/) 
e.  om )  ->  ( U `  b )  C_  ( U `  (/) ) )
106, 9mpan2 654 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  b )  C_  ( U `  (/) ) )
11 vex 2961 . . . . . . . . . 10  |-  t  e. 
_V
1211rnex 5136 . . . . . . . . 9  |-  ran  t  e.  _V
1312uniex 4708 . . . . . . . 8  |-  U. ran  t  e.  _V
142seqom0g 6716 . . . . . . . 8  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1513, 14ax-mp 5 . . . . . . 7  |-  ( U `
 (/) )  =  U. ran  t
1610, 15syl6sseq 3396 . . . . . 6  |-  ( b  e.  om  ->  ( U `  b )  C_ 
U. ran  t )
17 sseq1 3371 . . . . . 6  |-  ( ( U `  b )  =  a  ->  (
( U `  b
)  C_  U. ran  t  <->  a 
C_  U. ran  t ) )
1816, 17syl5ibcom 213 . . . . 5  |-  ( b  e.  om  ->  (
( U `  b
)  =  a  -> 
a  C_  U. ran  t
) )
1918rexlimiv 2826 . . . 4  |-  ( E. b  e.  om  ( U `  b )  =  a  ->  a  C_  U.
ran  t )
205, 19sylbi 189 . . 3  |-  ( a  e.  ran  U  -> 
a  C_  U. ran  t
)
211, 20mprgbir 2778 . 2  |-  U. ran  U 
C_  U. ran  t
22 fnfvelrn 5870 . . . . 5  |-  ( ( U  Fn  om  /\  (/) 
e.  om )  ->  ( U `  (/) )  e. 
ran  U )
233, 6, 22mp2an 655 . . . 4  |-  ( U `
 (/) )  e.  ran  U
2415, 23eqeltrri 2509 . . 3  |-  U. ran  t  e.  ran  U
25 elssuni 4045 . . 3  |-  ( U. ran  t  e.  ran  U  ->  U. ran  t  C_  U.
ran  U )
2624, 25ax-mp 5 . 2  |-  U. ran  t  C_  U. ran  U
2721, 26eqssi 3366 1  |-  U. ran  U  =  U. ran  t
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   U.cuni 4017   omcom 4848   ran crn 4882    Fn wfn 5452   ` cfv 5457    e. cmpt2 6086  seq𝜔cseqom 6707
This theorem is referenced by:  fin23lem17  8223  fin23lem31  8228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-recs 6636  df-rdg 6671  df-seqom 6708
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