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Theorem hof1 15845
Description: The object part of the Hom functor 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 )
hof1.b  |-  B  =  ( Base `  C
)
hof1.h  |-  H  =  ( Hom  `  C
)
hof1.x  |-  ( ph  ->  X  e.  B )
hof1.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
hof1  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X H Y ) )

Proof of Theorem hof1
StepHypRef Expression
1 hofval.m . . . 4  |-  M  =  (HomF
`  C )
2 hofval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
31, 2hof1fval 15844 . . 3  |-  ( ph  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )
43oveqd 6294 . 2  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X ( Hom f  `  C ) Y ) )
5 eqid 2402 . . 3  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
6 hof1.b . . 3  |-  B  =  ( Base `  C
)
7 hof1.h . . 3  |-  H  =  ( Hom  `  C
)
8 hof1.x . . 3  |-  ( ph  ->  X  e.  B )
9 hof1.y . . 3  |-  ( ph  ->  Y  e.  B )
105, 6, 7, 8, 9homfval 15303 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
114, 10eqtrd 2443 1  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X H Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   1stc1st 6781   Basecbs 14839   Hom chom 14918   Catccat 15276   Hom f chomf 15278  HomFchof 15839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-homf 15282  df-hof 15841
This theorem is referenced by:  yon11  15855  yonedalem21  15864
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