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Theorem homahom 14194
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 14191 . . 3  |-  Rel  ( X H Y )
3 1st2ndbr 6396 . . 3  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
42, 3mpan 652 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
5 homahom.j . . 3  |-  J  =  (  Hom  `  C
)
61, 5homahom2 14193 . 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 set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4212   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    Hom chom 13540  Homachoma 14178
This theorem is referenced by:  arwhom  14206  coahom  14225  arwlid  14227  arwrid  14228  arwass  14229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-homa 14181
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