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Theorem homahom 14899
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 14896 . . 3  |-  Rel  ( X H Y )
3 1st2ndbr 6618 . . 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 14898 . 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 1369    e. wcel 1756   class class class wbr 4287   Rel wrel 4840   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   Hom chom 14241  Homachoma 14883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-1st 6572  df-2nd 6573  df-homa 14886
This theorem is referenced by:  arwhom  14911  coahom  14930  arwlid  14932  arwrid  14933  arwass  14934
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