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Theorem rcmob 25852
Description: The range of  ( cod_ `  T ) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
Hypotheses
Ref Expression
rcmob.1  |-  O  =  dom  ( id_ `  T
)
rcmob.2  |-  C  =  ( cod_ `  T
)
Assertion
Ref Expression
rcmob  |-  ( T  e.  Ded  ->  ran  C  =  O )

Proof of Theorem rcmob
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  ( dom_ `  T )  =  dom  ( dom_ `  T
)
3 eqid 2296 . . . 4  |-  ( dom_ `  T )  =  (
dom_ `  T )
4 rcmob.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
5 eqid 2296 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
6 rcmob.2 . . . 4  |-  C  =  ( cod_ `  T
)
72, 3, 4, 5, 6coda 25832 . . 3  |-  ( T  e.  Alg  ->  C : dom  ( dom_ `  T
) --> O )
8 frn 5411 . . 3  |-  ( C : dom  ( dom_ `  T ) --> O  ->  ran  C  C_  O )
91, 7, 83syl 18 . 2  |-  ( T  e.  Ded  ->  ran  C 
C_  O )
104, 3, 5, 6idosd 25847 . . . . . 6  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( ( ( dom_ `  T ) `  (
( id_ `  T
) `  a )
)  =  a  /\  ( C `  ( ( id_ `  T ) `
 a ) )  =  a ) )
1110simprd 449 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( C `  (
( id_ `  T
) `  a )
)  =  a )
12 ffun 5407 . . . . . . . 8  |-  ( C : dom  ( dom_ `  T ) --> O  ->  Fun  C )
131, 7, 123syl 18 . . . . . . 7  |-  ( T  e.  Ded  ->  Fun  C )
1413adantr 451 . . . . . 6  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  Fun  C )
152, 3, 4, 5idmoa 25834 . . . . . . . 8  |-  ( ( T  e.  Alg  /\  a  e.  O )  ->  ( ( id_ `  T
) `  a )  e.  dom  ( dom_ `  T
) )
163, 6dcsda 25836 . . . . . . . . . 10  |-  ( T  e.  Alg  ->  dom  ( dom_ `  T )  =  dom  C )
1716eqcomd 2301 . . . . . . . . 9  |-  ( T  e.  Alg  ->  dom  C  =  dom  ( dom_ `  T ) )
1817adantr 451 . . . . . . . 8  |-  ( ( T  e.  Alg  /\  a  e.  O )  ->  dom  C  =  dom  ( dom_ `  T )
)
1915, 18eleqtrrd 2373 . . . . . . 7  |-  ( ( T  e.  Alg  /\  a  e.  O )  ->  ( ( id_ `  T
) `  a )  e.  dom  C )
201, 19sylan 457 . . . . . 6  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( ( id_ `  T
) `  a )  e.  dom  C )
21 fvelrn 5677 . . . . . 6  |-  ( ( Fun  C  /\  (
( id_ `  T
) `  a )  e.  dom  C )  -> 
( C `  (
( id_ `  T
) `  a )
)  e.  ran  C
)
2214, 20, 21syl2anc 642 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  ( C `  (
( id_ `  T
) `  a )
)  e.  ran  C
)
2311, 22eqeltrrd 2371 . . . 4  |-  ( ( T  e.  Ded  /\  a  e.  O )  ->  a  e.  ran  C
)
2423ex 423 . . 3  |-  ( T  e.  Ded  ->  (
a  e.  O  -> 
a  e.  ran  C
) )
2524ssrdv 3198 . 2  |-  ( T  e.  Ded  ->  O  C_ 
ran  C )
269, 25eqssd 3209 1  |-  ( T  e.  Ded  ->  ran  C  =  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 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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