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Theorem ishomc 25789
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 25788 . . 3  |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { x  e.  M  |  ( ( D `
 x )  =  A  /\  ( C `
 x )  =  B ) } )
87eleq2d 2350 . 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 5525 . . . . . 6  |-  ( x  =  F  ->  ( D `  x )  =  ( D `  F ) )
109eqeq1d 2291 . . . . 5  |-  ( x  =  F  ->  (
( D `  x
)  =  A  <->  ( D `  F )  =  A ) )
11 fveq2 5525 . . . . . 6  |-  ( x  =  F  ->  ( C `  x )  =  ( C `  F ) )
1211eqeq1d 2291 . . . . 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 2923 . . 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 1623    e. wcel 1684   {crab 2547   <.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:  ishomd  25790
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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