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Theorem hof1fval 16090
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 2420 . . 3  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2420 . . 3  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2420 . . 3  |-  (comp `  C )  =  (comp `  C )
61, 2, 3, 4, 5hofval 16089 . 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 5882 . . 3  |-  ( Hom f  `  C )  e.  _V
8 fvex 5882 . . . . 5  |-  ( Base `  C )  e.  _V
98, 8xpex 6600 . . . 4  |-  ( (
Base `  C )  X.  ( Base `  C
) )  e.  _V
109, 9mpt2ex 6875 . . 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 6808 . 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 17 1  |-  ( ph  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   <.cop 3999    |-> cmpt 4475    X. cxp 4843   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6796   2ndc2nd 6797   Basecbs 15081   Hom chom 15161  compcco 15162   Catccat 15522   Hom f chomf 15524  HomFchof 16085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-hof 16087
This theorem is referenced by:  hof1  16091
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