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

Theorem homahom 15217
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 15214 . . 3  |-  Rel  ( X H Y )
3 1st2ndbr 6830 . . 3  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
42, 3mpan 670 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
5 homahom.j . . 3  |-  J  =  ( Hom  `  C
)
61, 5homahom2 15216 . 2  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 2nd `  F )  e.  ( X J Y ) )
74, 6syl 16 1  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  F )  e.  ( X J Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   class class class wbr 4447   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Hom chom 14559  Homachoma 15201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-1st 6781  df-2nd 6782  df-homa 15204
This theorem is referenced by:  arwhom  15229  coahom  15248  arwlid  15250  arwrid  15251  arwass  15252
  Copyright terms: Public domain W3C validator