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Theorem ishomb 25788
Description: The homset  ( ( hom `  T ) `
 <. A ,  B >. ). (Contributed by FL, 18-May-2007.)
Hypotheses
Ref Expression
ishomb.1  |-  O  =  dom  ( id_ `  T
)
ishomb.2  |-  M  =  dom  ( dom_ `  T
)
ishomb.3  |-  D  =  ( dom_ `  T
)
ishomb.4  |-  C  =  ( cod_ `  T
)
ishomb.5  |-  H  =  ( hom `  T
)
ishomb.6  |-  T  e. 
Cat OLD
Assertion
Ref Expression
ishomb  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { f  e.  M  |  ( ( D `
 f )  =  A  /\  ( C `
 f )  =  B ) } )
Distinct variable groups:    A, f    B, f    f, M    T, f
Allowed substitution hints:    C( f)    D( f)    H( f)    O( f)

Proof of Theorem ishomb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 5861 . 2  |-  ( A H B )  =  ( H `  <. A ,  B >. )
2 eqeq2 2292 . . . . 5  |-  ( x  =  A  ->  (
( D `  f
)  =  x  <->  ( D `  f )  =  A ) )
3 eqeq2 2292 . . . . 5  |-  ( y  =  B  ->  (
( C `  f
)  =  y  <->  ( C `  f )  =  B ) )
42, 3bi2anan9 843 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( D `
 f )  =  x  /\  ( C `
 f )  =  y )  <->  ( ( D `  f )  =  A  /\  ( C `  f )  =  B ) ) )
54rabbidv 2780 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  { f  e.  M  |  ( ( D `
 f )  =  x  /\  ( C `
 f )  =  y ) }  =  { f  e.  M  |  ( ( D `
 f )  =  A  /\  ( C `
 f )  =  B ) } )
6 ishomb.5 . . . 4  |-  H  =  ( hom `  T
)
7 ishomb.6 . . . . 5  |-  T  e. 
Cat OLD
8 ishomb.1 . . . . . 6  |-  O  =  dom  ( id_ `  T
)
9 ishomb.2 . . . . . 6  |-  M  =  dom  ( dom_ `  T
)
10 ishomb.3 . . . . . 6  |-  D  =  ( dom_ `  T
)
11 ishomb.4 . . . . . 6  |-  C  =  ( cod_ `  T
)
128, 9, 10, 11ishoma 25787 . . . . 5  |-  ( T  e.  Cat OLD  ->  ( hom `  T )  =  ( x  e.  O ,  y  e.  O  |->  { f  e.  M  |  ( ( D `  f )  =  x  /\  ( C `  f )  =  y ) } ) )
137, 12ax-mp 8 . . . 4  |-  ( hom `  T )  =  ( x  e.  O , 
y  e.  O  |->  { f  e.  M  | 
( ( D `  f )  =  x  /\  ( C `  f )  =  y ) } )
146, 13eqtri 2303 . . 3  |-  H  =  ( x  e.  O ,  y  e.  O  |->  { f  e.  M  |  ( ( D `
 f )  =  x  /\  ( C `
 f )  =  y ) } )
15 fvex 5539 . . . . . 6  |-  ( dom_ `  T )  e.  _V
1615dmex 4941 . . . . 5  |-  dom  ( dom_ `  T )  e. 
_V
179, 16eqeltri 2353 . . . 4  |-  M  e. 
_V
1817rabex 4165 . . 3  |-  { f  e.  M  |  ( ( D `  f
)  =  A  /\  ( C `  f )  =  B ) }  e.  _V
195, 14, 18ovmpt2a 5978 . 2  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( A H B )  =  { f  e.  M  |  ( ( D `  f
)  =  A  /\  ( C `  f )  =  B ) } )
201, 19syl5eqr 2329 1  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { f  e.  M  |  ( ( D `
 f )  =  A  /\  ( C `
 f )  =  B ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   <.cop 3643   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714    Cat
OLD ccatOLD 25752   homchomOLD 25785
This theorem is referenced by:  ishomc  25789
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-homOLD 25786
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