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Theorem rdmob 25748
Description: The range of  ( dom_ `  T ) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
Hypotheses
Ref Expression
rdmob.1  |-  O  =  dom  ( id_ `  T
)
rdmob.2  |-  D  =  ( dom_ `  T
)
Assertion
Ref Expression
rdmob  |-  ( T  e.  Ded  ->  ran  D  =  O )

Proof of Theorem rdmob
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dedalg 25743 . . 3  |-  ( T  e.  Ded  ->  T  e.  Alg  )
2 eqid 2283 . . . 4  |-  dom  D  =  dom  D
3 rdmob.2 . . . 4  |-  D  =  ( dom_ `  T
)
4 rdmob.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
5 eqid 2283 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
62, 3, 4, 5doma 25728 . . 3  |-  ( T  e.  Alg  ->  D : dom  D --> O )
7 frn 5395 . . 3  |-  ( D : dom  D --> O  ->  ran  D  C_  O )
81, 6, 73syl 18 . 2  |-  ( T  e.  Ded  ->  ran  D 
C_  O )
9 eqid 2283 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
104, 3, 5, 9idosd 25744 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( ( D `  ( ( id_ `  T
) `  a )
)  =  a  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  a ) )  =  a ) )
11 eqid 2283 . . . . . . . . 9  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
12 eqid 2283 . . . . . . . . 9  |-  ( dom_ `  T )  =  (
dom_ `  T )
1311, 12, 4, 5ida 25730 . . . . . . . 8  |-  ( T  e.  Alg  ->  ( id_ `  T ) : O --> dom  ( dom_ `  T ) )
14 ffvelrn 5663 . . . . . . . . 9  |-  ( ( ( id_ `  T
) : O --> dom  ( dom_ `  T )  /\  a  e.  O )  ->  ( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
) )
1514ex 423 . . . . . . . 8  |-  ( ( id_ `  T ) : O --> dom  ( dom_ `  T )  -> 
( a  e.  O  ->  ( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
) ) )
161, 13, 153syl 18 . . . . . . 7  |-  ( T  e.  Ded  ->  (
a  e.  O  -> 
( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
) ) )
17 eqid 2283 . . . . . . . . 9  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
1811, 12, 17, 5doma 25728 . . . . . . . 8  |-  ( T  e.  Alg  ->  ( dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T ) )
19 ffun 5391 . . . . . . . . . 10  |-  ( (
dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  Fun  ( dom_ `  T )
)
20 fvelrn 5661 . . . . . . . . . . 11  |-  ( ( Fun  ( dom_ `  T
)  /\  ( ( id_ `  T ) `  a )  e.  dom  ( dom_ `  T )
)  ->  ( ( dom_ `  T ) `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) )
2120ex 423 . . . . . . . . . 10  |-  ( Fun  ( dom_ `  T
)  ->  ( (
( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
)  ->  ( ( dom_ `  T ) `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) ) )
2219, 21syl 15 . . . . . . . . 9  |-  ( (
dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  (
( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
)  ->  ( ( dom_ `  T ) `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) ) )
233fveq1i 5526 . . . . . . . . . 10  |-  ( D `
 ( ( id_ `  T ) `  a
) )  =  ( ( dom_ `  T
) `  ( ( id_ `  T ) `  a ) )
2423eleq1i 2346 . . . . . . . . 9  |-  ( ( D `  ( ( id_ `  T ) `
 a ) )  e.  ran  ( dom_ `  T )  <->  ( ( dom_ `  T ) `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) )
2522, 24syl6ibr 218 . . . . . . . 8  |-  ( (
dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  (
( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
)  ->  ( D `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) ) )
261, 18, 253syl 18 . . . . . . 7  |-  ( T  e.  Ded  ->  (
( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
)  ->  ( D `  ( ( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) ) )
2716, 26syld 40 . . . . . 6  |-  ( T  e.  Ded  ->  (
a  e.  O  -> 
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) ) )
2827imp 418 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T ) )
29 eleq1 2343 . . . . . . . 8  |-  ( ( D `  ( ( id_ `  T ) `
 a ) )  =  a  ->  (
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T )  <->  a  e.  ran  ( dom_ `  T
) ) )
303rneqi 4905 . . . . . . . . 9  |-  ran  D  =  ran  ( dom_ `  T
)
3130eleq2i 2347 . . . . . . . 8  |-  ( a  e.  ran  D  <->  a  e.  ran  ( dom_ `  T
) )
3229, 31syl6bbr 254 . . . . . . 7  |-  ( ( D `  ( ( id_ `  T ) `
 a ) )  =  a  ->  (
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T )  <->  a  e.  ran  D ) )
3332biimpd 198 . . . . . 6  |-  ( ( D `  ( ( id_ `  T ) `
 a ) )  =  a  ->  (
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T )  -> 
a  e.  ran  D
) )
3433adantr 451 . . . . 5  |-  ( ( ( D `  (
( id_ `  T
) `  a )
)  =  a  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  a ) )  =  a )  ->  (
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T )  -> 
a  e.  ran  D
) )
3510, 28, 34sylc 56 . . . 4  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  a  e.  ran  D
)
3635ex 423 . . 3  |-  ( T  e.  Ded  ->  (
a  e.  O  -> 
a  e.  ran  D
) )
3736ssrdv 3185 . 2  |-  ( T  e.  Ded  ->  O  C_ 
ran  D )
388, 37eqssd 3196 1  |-  ( T  e.  Ded  ->  ran  D  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689   ran crn 4690   Fun wfun 5249   -->wf 5251   ` cfv 5255    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   Dedcded 25734
This theorem is referenced by:  aidm2  25750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735
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