MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hof1 Structured version   Unicode version

Theorem hof1 15187
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 15186 . . 3  |-  ( ph  ->  ( 1st `  M
)  =  ( Hom f  `  C ) )
43oveqd 6220 . 2  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X ( Hom f  `  C ) Y ) )
5 eqid 2454 . . 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 14754 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
114, 10eqtrd 2495 1  |-  ( ph  ->  ( X ( 1st `  M ) Y )  =  ( X H Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   1stc1st 6688   Basecbs 14296   Hom chom 14372   Catccat 14725   Hom f chomf 14727  HomFchof 15181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-homf 14731  df-hof 15183
This theorem is referenced by:  yon11  15197  yonedalem21  15206
  Copyright terms: Public domain W3C validator