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Theorem hof1 14351
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 14350 . . 3  |-  ( ph  ->  ( 1st `  M
)  =  (  Homf  `  C ) )
43oveqd 6098 . 2  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X (  Homf 
`  C ) Y ) )
5 eqid 2436 . . 3  |-  (  Homf  `  C )  =  (  Homf 
`  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 13918 . 2  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
114, 10eqtrd 2468 1  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   1stc1st 6347   Basecbs 13469    Hom chom 13540   Catccat 13889    Homf chomf 13891  HomFchof 14345
This theorem is referenced by:  yon11  14361  yonedalem21  14370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-homf 13895  df-hof 14347
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