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Theorem homahom 15025
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 15022 . . 3  |-  Rel  ( X H Y )
3 1st2ndbr 6732 . . 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 15024 . 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 1370    e. wcel 1758   class class class wbr 4399   Rel wrel 4952   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685   Hom chom 14367  Homachoma 15009
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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-1st 6686  df-2nd 6687  df-homa 15012
This theorem is referenced by:  arwhom  15037  coahom  15056  arwlid  15058  arwrid  15059  arwass  15060
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