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Theorem hof1fval 15383
Description: The object part of the Hom functor is the  Hom f operation, which is just a functionalized version of  Hom. That is, it is a two argument function, which maps  X ,  Y to the set of morphisms from  X to  Y. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m  |-  M  =  (HomF
`  C )
hofval.c  |-  ( ph  ->  C  e.  Cat )
Assertion
Ref Expression
hof1fval  |-  ( ph  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )

Proof of Theorem hof1fval
Dummy variables  f 
g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3  |-  M  =  (HomF
`  C )
2 hofval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 eqid 2467 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2467 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2467 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5hofval 15382 . 2  |-  ( ph  ->  M  =  <. ( Hom f  `  C ) ,  ( x  e.  ( (
Base `  C )  X.  ( Base `  C
) ) ,  y  e.  ( ( Base `  C )  X.  ( Base `  C ) ) 
|->  ( f  e.  ( ( 1st `  y
) ( Hom  `  C
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  C ) ( 2nd `  y ) )  |->  ( h  e.  ( ( Hom  `  C ) `  x )  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) ) >.
)
7 fvex 5876 . . 3  |-  ( Hom f  `  C )  e.  _V
8 fvex 5876 . . . . 5  |-  ( Base `  C )  e.  _V
98, 8xpex 6589 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  C
) )  e.  _V
109, 9mpt2ex 6861 . . 3  |-  ( x  e.  ( ( Base `  C )  X.  ( Base `  C ) ) ,  y  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  |->  ( f  e.  ( ( 1st `  y ) ( Hom  `  C
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  C ) ( 2nd `  y ) )  |->  ( h  e.  ( ( Hom  `  C ) `  x )  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) )  e. 
_V
117, 10op1std 6795 . 2  |-  ( M  =  <. ( Hom f  `  C ) ,  ( x  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) ,  y  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |->  ( f  e.  ( ( 1st `  y ) ( Hom  `  C ) ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  C
) ( 2nd `  y
) )  |->  ( h  e.  ( ( Hom  `  C ) `  x
)  |->  ( ( g ( x (comp `  C ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  C
) ( 2nd `  y
) ) f ) ) ) ) >.  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )
126, 11syl 16 1  |-  ( ph  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   <.cop 4033    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784   Basecbs 14493   Hom chom 14569  compcco 14570   Catccat 14922   Hom f chomf 14924  HomFchof 15378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-hof 15380
This theorem is referenced by:  hof1  15384
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