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Theorem cehm 25793
Description: Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)
Hypotheses
Ref Expression
cehm.1  |-  O  =  dom  ( id_ `  T
)
cehm.2  |-  C  =  ( cod_ `  T
)
cehm.5  |-  H  =  ( hom `  T
)
Assertion
Ref Expression
cehm  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( C `  F )  =  B ) )

Proof of Theorem cehm
StepHypRef Expression
1 cehm.1 . . . . 5  |-  O  =  dom  ( id_ `  T
)
2 eqid 2283 . . . . 5  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
3 eqid 2283 . . . . 5  |-  ( dom_ `  T )  =  (
dom_ `  T )
4 cehm.2 . . . . 5  |-  C  =  ( cod_ `  T
)
5 cehm.5 . . . . 5  |-  H  =  ( hom `  T
)
61, 2, 3, 4, 5ishomd 25790 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  <->  ( F  e.  dom  ( dom_ `  T
)  /\  ( ( dom_ `  T ) `  F )  =  A  /\  ( C `  F )  =  B ) ) )
76biimpa 470 . . 3  |-  ( ( ( T  e.  Cat OLD 
/\  A  e.  O  /\  B  e.  O
)  /\  F  e.  ( H `  <. A ,  B >. ) )  -> 
( F  e.  dom  ( dom_ `  T )  /\  ( ( dom_ `  T
) `  F )  =  A  /\  ( C `  F )  =  B ) )
87simp3d 969 . 2  |-  ( ( ( T  e.  Cat OLD 
/\  A  e.  O  /\  B  e.  O
)  /\  F  e.  ( H `  <. A ,  B >. ) )  -> 
( C `  F
)  =  B )
98ex 423 1  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( C `  F )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   dom cdm 4689   ` cfv 5255   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714    Cat
OLD ccatOLD 25752   homchomOLD 25785
This theorem is referenced by:  cmphmia  25798  cmpassoh  25801  homgrf  25802  idmon  25817  isepic  25824
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-alg 25716  df-ded 25735  df-catOLD 25753  df-homOLD 25786
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