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Theorem ishomb 25891
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 5877 . 2  |-  ( A H B )  =  ( H `  <. A ,  B >. )
2 eqeq2 2305 . . . . 5  |-  ( x  =  A  ->  (
( D `  f
)  =  x  <->  ( D `  f )  =  A ) )
3 eqeq2 2305 . . . . 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 2793 . . 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 25890 . . . . 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 2316 . . 3  |-  H  =  ( x  e.  O ,  y  e.  O  |->  { f  e.  M  |  ( ( D `
 f )  =  x  /\  ( C `
 f )  =  y ) } )
15 fvex 5555 . . . . . 6  |-  ( dom_ `  T )  e.  _V
1615dmex 4957 . . . . 5  |-  dom  ( dom_ `  T )  e. 
_V
179, 16eqeltri 2366 . . . 4  |-  M  e. 
_V
1817rabex 4181 . . 3  |-  { f  e.  M  |  ( ( D `  f
)  =  A  /\  ( C `  f )  =  B ) }  e.  _V
195, 14, 18ovmpt2a 5994 . 2  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( A H B )  =  { f  e.  M  |  ( ( D `  f
)  =  A  /\  ( C `  f )  =  B ) } )
201, 19syl5eqr 2342 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817    Cat
OLD ccatOLD 25855   homchomOLD 25888
This theorem is referenced by:  ishomc  25892
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-homOLD 25889
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