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Theorem arwhom 14924
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
arwrcl.a  |-  A  =  (Nat `  C )
arwhom.j  |-  J  =  ( Hom  `  C
)
Assertion
Ref Expression
arwhom  |-  ( F  e.  A  ->  ( 2nd `  F )  e.  ( (domA `  F ) J (coda `  F ) ) )

Proof of Theorem arwhom
StepHypRef Expression
1 arwrcl.a . . 3  |-  A  =  (Nat `  C )
2 eqid 2443 . . 3  |-  (Homa `  C
)  =  (Homa `  C
)
31, 2arwhoma 14918 . 2  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) (Homa `  C
) (coda
`  F ) ) )
4 arwhom.j . . 3  |-  J  =  ( Hom  `  C
)
52, 4homahom 14912 . 2  |-  ( F  e.  ( (domA `  F ) (Homa
`  C ) (coda `  F ) )  -> 
( 2nd `  F
)  e.  ( (domA `  F
) J (coda `  F
) ) )
63, 5syl 16 1  |-  ( F  e.  A  ->  ( 2nd `  F )  e.  ( (domA `  F ) J (coda `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   2ndc2nd 6581   Hom chom 14254  domAcdoma 14893  codaccoda 14894  Natcarw 14895  Homachoma 14896
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-1st 6582  df-2nd 6583  df-doma 14897  df-coda 14898  df-homa 14899  df-arw 14900
This theorem is referenced by: (None)
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