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Theorem fin23lem16 8727
Description: Lemma for fin23 8781. 
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 4283 . . 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 7132 . . . . 5  |-  U  Fn  om
4 fvelrnb 5921 . . . . 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 6714 . . . . . . . 8  |-  (/)  e.  om
7 0ss 3819 . . . . . . . . 9  |-  (/)  C_  b
82fin23lem15 8726 . . . . . . . . 9  |-  ( ( ( b  e.  om  /\  (/)  e.  om )  /\  (/)  C_  b )  ->  ( U `  b )  C_  ( U `  (/) ) )
97, 8mpan2 671 . . . . . . . 8  |-  ( ( b  e.  om  /\  (/) 
e.  om )  ->  ( U `  b )  C_  ( U `  (/) ) )
106, 9mpan2 671 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  b )  C_  ( U `  (/) ) )
11 vex 3121 . . . . . . . . . 10  |-  t  e. 
_V
1211rnex 6729 . . . . . . . . 9  |-  ran  t  e.  _V
1312uniex 6591 . . . . . . . 8  |-  U. ran  t  e.  _V
142seqom0g 7133 . . . . . . . 8  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1513, 14ax-mp 5 . . . . . . 7  |-  ( U `
 (/) )  =  U. ran  t
1610, 15syl6sseq 3555 . . . . . 6  |-  ( b  e.  om  ->  ( U `  b )  C_ 
U. ran  t )
17 sseq1 3530 . . . . . 6  |-  ( ( U `  b )  =  a  ->  (
( U `  b
)  C_  U. ran  t  <->  a 
C_  U. ran  t ) )
1816, 17syl5ibcom 220 . . . . 5  |-  ( b  e.  om  ->  (
( U `  b
)  =  a  -> 
a  C_  U. ran  t
) )
1918rexlimiv 2953 . . . 4  |-  ( E. b  e.  om  ( U `  b )  =  a  ->  a  C_  U.
ran  t )
205, 19sylbi 195 . . 3  |-  ( a  e.  ran  U  -> 
a  C_  U. ran  t
)
211, 20mprgbir 2831 . 2  |-  U. ran  U 
C_  U. ran  t
22 fnfvelrn 6029 . . . . 5  |-  ( ( U  Fn  om  /\  (/) 
e.  om )  ->  ( U `  (/) )  e. 
ran  U )
233, 6, 22mp2an 672 . . . 4  |-  ( U `
 (/) )  e.  ran  U
2415, 23eqeltrri 2552 . . 3  |-  U. ran  t  e.  ran  U
25 elssuni 4281 . . 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 3525 1  |-  U. ran  U  =  U. ran  t
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ifcif 3945   U.cuni 4251   ran crn 5006    Fn wfn 5589   ` cfv 5594    |-> cmpt2 6297   omcom 6695  seq𝜔cseqom 7124
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-seqom 7125
This theorem is referenced by:  fin23lem17  8730  fin23lem31  8735
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