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Theorem fin23lem16 8765
Description: Lemma for fin23 8819. 
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 4247 . . 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 7176 . . . . 5  |-  U  Fn  om
4 fvelrnb 5924 . . . . 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 6722 . . . . . . . 8  |-  (/)  e.  om
7 0ss 3791 . . . . . . . . 9  |-  (/)  C_  b
82fin23lem15 8764 . . . . . . . . 9  |-  ( ( ( b  e.  om  /\  (/)  e.  om )  /\  (/)  C_  b )  ->  ( U `  b )  C_  ( U `  (/) ) )
97, 8mpan2 675 . . . . . . . 8  |-  ( ( b  e.  om  /\  (/) 
e.  om )  ->  ( U `  b )  C_  ( U `  (/) ) )
106, 9mpan2 675 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  b )  C_  ( U `  (/) ) )
11 vex 3084 . . . . . . . . . 10  |-  t  e. 
_V
1211rnex 6737 . . . . . . . . 9  |-  ran  t  e.  _V
1312uniex 6597 . . . . . . . 8  |-  U. ran  t  e.  _V
142seqom0g 7177 . . . . . . . 8  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1513, 14ax-mp 5 . . . . . . 7  |-  ( U `
 (/) )  =  U. ran  t
1610, 15syl6sseq 3510 . . . . . 6  |-  ( b  e.  om  ->  ( U `  b )  C_ 
U. ran  t )
17 sseq1 3485 . . . . . 6  |-  ( ( U `  b )  =  a  ->  (
( U `  b
)  C_  U. ran  t  <->  a 
C_  U. ran  t ) )
1816, 17syl5ibcom 223 . . . . 5  |-  ( b  e.  om  ->  (
( U `  b
)  =  a  -> 
a  C_  U. ran  t
) )
1918rexlimiv 2911 . . . 4  |-  ( E. b  e.  om  ( U `  b )  =  a  ->  a  C_  U.
ran  t )
205, 19sylbi 198 . . 3  |-  ( a  e.  ran  U  -> 
a  C_  U. ran  t
)
211, 20mprgbir 2789 . 2  |-  U. ran  U 
C_  U. ran  t
22 fnfvelrn 6030 . . . . 5  |-  ( ( U  Fn  om  /\  (/) 
e.  om )  ->  ( U `  (/) )  e. 
ran  U )
233, 6, 22mp2an 676 . . . 4  |-  ( U `
 (/) )  e.  ran  U
2415, 23eqeltrri 2507 . . 3  |-  U. ran  t  e.  ran  U
25 elssuni 4245 . . 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 3480 1  |-  U. ran  U  =  U. ran  t
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   E.wrex 2776   _Vcvv 3081    i^i cin 3435    C_ wss 3436   (/)c0 3761   ifcif 3909   U.cuni 4216   ran crn 4850    Fn wfn 5592   ` cfv 5597    |-> cmpt2 6303   omcom 6702  seq𝜔cseqom 7168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-seqom 7169
This theorem is referenced by:  fin23lem17  8768  fin23lem31  8773
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