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

Theorem homahom 13871
Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homahom.h  |-  H  =  (Homa
`  C )
homahom.j  |-  J  =  (  Hom  `  C
)
Assertion
Ref Expression
homahom  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X J Y ) )

Proof of Theorem homahom
StepHypRef Expression
1 homahom.h . . . 4  |-  H  =  (Homa
`  C )
21homarel 13868 . . 3  |-  Rel  ( X H Y )
3 1st2ndbr 6169 . . 3  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
42, 3mpan 651 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
5 homahom.j . . 3  |-  J  =  (  Hom  `  C
)
61, 5homahom2 13870 . 2  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 2nd `  F )  e.  ( X J Y ) )
74, 6syl 15 1  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X J Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    Hom chom 13219  Homachoma 13855
This theorem is referenced by:  arwhom  13883  coahom  13902  arwlid  13904  arwrid  13905  arwass  13906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-homa 13858
  Copyright terms: Public domain W3C validator