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Theorem rdmob 25851
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 25846 . . 3  |-  ( T  e.  Ded  ->  T  e.  Alg  )
2 eqid 2296 . . . 4  |-  dom  D  =  dom  D
3 rdmob.2 . . . 4  |-  D  =  ( dom_ `  T
)
4 rdmob.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
5 eqid 2296 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
62, 3, 4, 5doma 25831 . . 3  |-  ( T  e.  Alg  ->  D : dom  D --> O )
7 frn 5411 . . 3  |-  ( D : dom  D --> O  ->  ran  D  C_  O )
81, 6, 73syl 18 . 2  |-  ( T  e.  Ded  ->  ran  D 
C_  O )
9 eqid 2296 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
104, 3, 5, 9idosd 25847 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( ( D `  ( ( id_ `  T
) `  a )
)  =  a  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  a ) )  =  a ) )
11 eqid 2296 . . . . . . . . 9  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
12 eqid 2296 . . . . . . . . 9  |-  ( dom_ `  T )  =  (
dom_ `  T )
1311, 12, 4, 5ida 25833 . . . . . . . 8  |-  ( T  e.  Alg  ->  ( id_ `  T ) : O --> dom  ( dom_ `  T ) )
14 ffvelrn 5679 . . . . . . . . 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 2296 . . . . . . . . 9  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
1811, 12, 17, 5doma 25831 . . . . . . . 8  |-  ( T  e.  Alg  ->  ( dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T ) )
19 ffun 5407 . . . . . . . . . 10  |-  ( (
dom_ `  T ) : dom  ( dom_ `  T
) --> dom  ( id_ `  T )  ->  Fun  ( dom_ `  T )
)
20 fvelrn 5677 . . . . . . . . . . 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 5542 . . . . . . . . . 10  |-  ( D `
 ( ( id_ `  T ) `  a
) )  =  ( ( dom_ `  T
) `  ( ( id_ `  T ) `  a ) )
2423eleq1i 2359 . . . . . . . . 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 2356 . . . . . . . 8  |-  ( ( D `  ( ( id_ `  T ) `
 a ) )  =  a  ->  (
( D `  (
( id_ `  T
) `  a )
)  e.  ran  ( dom_ `  T )  <->  a  e.  ran  ( dom_ `  T
) ) )
303rneqi 4921 . . . . . . . . 9  |-  ran  D  =  ran  ( dom_ `  T
)
3130eleq2i 2360 . . . . . . . 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 3198 . 2  |-  ( T  e.  Ded  ->  O  C_ 
ran  D )
388, 37eqssd 3209 1  |-  ( T  e.  Ded  ->  ran  D  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   dom cdm 4705   ran crn 4706   Fun wfun 5265   -->wf 5267   ` cfv 5271    Alg calg 25814   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   Dedcded 25837
This theorem is referenced by:  aidm2  25853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-alg 25819  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838
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