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Theorem fin23lem16 8175
Description: Lemma for fin23 8229. 
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 4009 . . 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 6675 . . . . 5  |-  U  Fn  om
4 fvelrnb 5737 . . . . 5  |-  ( U  Fn  om  ->  (
a  e.  ran  U  <->  E. b  e.  om  ( U `  b )  =  a ) )
53, 4ax-mp 8 . . . 4  |-  ( a  e.  ran  U  <->  E. b  e.  om  ( U `  b )  =  a )
6 peano1 4827 . . . . . . . 8  |-  (/)  e.  om
7 0ss 3620 . . . . . . . . 9  |-  (/)  C_  b
82fin23lem15 8174 . . . . . . . . 9  |-  ( ( ( b  e.  om  /\  (/)  e.  om )  /\  (/)  C_  b )  ->  ( U `  b )  C_  ( U `  (/) ) )
97, 8mpan2 653 . . . . . . . 8  |-  ( ( b  e.  om  /\  (/) 
e.  om )  ->  ( U `  b )  C_  ( U `  (/) ) )
106, 9mpan2 653 . . . . . . 7  |-  ( b  e.  om  ->  ( U `  b )  C_  ( U `  (/) ) )
11 vex 2923 . . . . . . . . . 10  |-  t  e. 
_V
1211rnex 5096 . . . . . . . . 9  |-  ran  t  e.  _V
1312uniex 4668 . . . . . . . 8  |-  U. ran  t  e.  _V
142seqom0g 6676 . . . . . . . 8  |-  ( U. ran  t  e.  _V  ->  ( U `  (/) )  = 
U. ran  t )
1513, 14ax-mp 8 . . . . . . 7  |-  ( U `
 (/) )  =  U. ran  t
1610, 15syl6sseq 3358 . . . . . 6  |-  ( b  e.  om  ->  ( U `  b )  C_ 
U. ran  t )
17 sseq1 3333 . . . . . 6  |-  ( ( U `  b )  =  a  ->  (
( U `  b
)  C_  U. ran  t  <->  a 
C_  U. ran  t ) )
1816, 17syl5ibcom 212 . . . . 5  |-  ( b  e.  om  ->  (
( U `  b
)  =  a  -> 
a  C_  U. ran  t
) )
1918rexlimiv 2788 . . . 4  |-  ( E. b  e.  om  ( U `  b )  =  a  ->  a  C_  U.
ran  t )
205, 19sylbi 188 . . 3  |-  ( a  e.  ran  U  -> 
a  C_  U. ran  t
)
211, 20mprgbir 2740 . 2  |-  U. ran  U 
C_  U. ran  t
22 fnfvelrn 5830 . . . . 5  |-  ( ( U  Fn  om  /\  (/) 
e.  om )  ->  ( U `  (/) )  e. 
ran  U )
233, 6, 22mp2an 654 . . . 4  |-  ( U `
 (/) )  e.  ran  U
2415, 23eqeltrri 2479 . . 3  |-  U. ran  t  e.  ran  U
25 elssuni 4007 . . 3  |-  ( U. ran  t  e.  ran  U  ->  U. ran  t  C_  U.
ran  U )
2624, 25ax-mp 8 . 2  |-  U. ran  t  C_  U. ran  U
2721, 26eqssi 3328 1  |-  U. ran  U  =  U. ran  t
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2671   _Vcvv 2920    i^i cin 3283    C_ wss 3284   (/)c0 3592   ifcif 3703   U.cuni 3979   omcom 4808   ran crn 4842    Fn wfn 5412   ` cfv 5417    e. cmpt2 6046  seq𝜔cseqom 6667
This theorem is referenced by:  fin23lem17  8178  fin23lem31  8183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-recs 6596  df-rdg 6631  df-seqom 6668
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