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Theorem fin23lem16 8496
Description: Lemma for fin23 8550. 
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 4118 . . 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 6902 . . . . 5  |-  U  Fn  om
4 fvelrnb 5734 . . . . 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 6490 . . . . . . . 8  |-  (/)  e.  om
7 0ss 3661 . . . . . . . . 9  |-  (/)  C_  b
82fin23lem15 8495 . . . . . . . . 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 2970 . . . . . . . . . 10  |-  t  e. 
_V
1211rnex 6507 . . . . . . . . 9  |-  ran  t  e.  _V
1312uniex 6371 . . . . . . . 8  |-  U. ran  t  e.  _V
142seqom0g 6903 . . . . . . . 8  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1513, 14ax-mp 5 . . . . . . 7  |-  ( U `
 (/) )  =  U. ran  t
1610, 15syl6sseq 3397 . . . . . 6  |-  ( b  e.  om  ->  ( U `  b )  C_ 
U. ran  t )
17 sseq1 3372 . . . . . 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 2830 . . . 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 2781 . 2  |-  U. ran  U 
C_  U. ran  t
22 fnfvelrn 5835 . . . . 5  |-  ( ( U  Fn  om  /\  (/) 
e.  om )  ->  ( U `  (/) )  e. 
ran  U )
233, 6, 22mp2an 672 . . . 4  |-  ( U `
 (/) )  e.  ran  U
2415, 23eqeltrri 2509 . . 3  |-  U. ran  t  e.  ran  U
25 elssuni 4116 . . 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 3367 1  |-  U. ran  U  =  U. ran  t
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   (/)c0 3632   ifcif 3786   U.cuni 4086   ran crn 4836    Fn wfn 5408   ` cfv 5413    e. cmpt2 6088   omcom 6471  seq𝜔cseqom 6894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-seqom 6895
This theorem is referenced by:  fin23lem17  8499  fin23lem31  8504
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