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Theorem ishomc 25892
Description: The predicate  F  e.  ( ( hom `  T
) `  <. A ,  B >. ) JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)
Hypotheses
Ref Expression
ishomc.1  |-  O  =  dom  ( id_ `  T
)
ishomc.2  |-  M  =  dom  ( dom_ `  T
)
ishomc.3  |-  D  =  ( dom_ `  T
)
ishomc.4  |-  C  =  ( cod_ `  T
)
ishomc.5  |-  H  =  ( hom `  T
)
ishomc.6  |-  T  e. 
Cat OLD
Assertion
Ref Expression
ishomc  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  <->  ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B ) ) )

Proof of Theorem ishomc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ishomc.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
2 ishomc.2 . . . 4  |-  M  =  dom  ( dom_ `  T
)
3 ishomc.3 . . . 4  |-  D  =  ( dom_ `  T
)
4 ishomc.4 . . . 4  |-  C  =  ( cod_ `  T
)
5 ishomc.5 . . . 4  |-  H  =  ( hom `  T
)
6 ishomc.6 . . . 4  |-  T  e. 
Cat OLD
71, 2, 3, 4, 5, 6ishomb 25891 . . 3  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { x  e.  M  |  ( ( D `
 x )  =  A  /\  ( C `
 x )  =  B ) } )
87eleq2d 2363 . 2  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  <->  F  e.  { x  e.  M  | 
( ( D `  x )  =  A  /\  ( C `  x )  =  B ) } ) )
9 fveq2 5541 . . . . . 6  |-  ( x  =  F  ->  ( D `  x )  =  ( D `  F ) )
109eqeq1d 2304 . . . . 5  |-  ( x  =  F  ->  (
( D `  x
)  =  A  <->  ( D `  F )  =  A ) )
11 fveq2 5541 . . . . . 6  |-  ( x  =  F  ->  ( C `  x )  =  ( C `  F ) )
1211eqeq1d 2304 . . . . 5  |-  ( x  =  F  ->  (
( C `  x
)  =  B  <->  ( C `  F )  =  B ) )
1310, 12anbi12d 691 . . . 4  |-  ( x  =  F  ->  (
( ( D `  x )  =  A  /\  ( C `  x )  =  B )  <->  ( ( D `
 F )  =  A  /\  ( C `
 F )  =  B ) ) )
1413elrab 2936 . . 3  |-  ( F  e.  { x  e.  M  |  ( ( D `  x )  =  A  /\  ( C `  x )  =  B ) }  <->  ( F  e.  M  /\  (
( D `  F
)  =  A  /\  ( C `  F )  =  B ) ) )
15 3anass 938 . . 3  |-  ( ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B )  <->  ( F  e.  M  /\  (
( D `  F
)  =  A  /\  ( C `  F )  =  B ) ) )
1614, 15bitr4i 243 . 2  |-  ( F  e.  { x  e.  M  |  ( ( D `  x )  =  A  /\  ( C `  x )  =  B ) }  <->  ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B ) )
178, 16syl6bb 252 1  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  <->  ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   <.cop 3656   dom cdm 4705   ` cfv 5271   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817    Cat
OLD ccatOLD 25855   homchomOLD 25888
This theorem is referenced by:  ishomd  25893
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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