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Theorem arwhom 15456
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 2457 . . 3  |-  (Homa `  C
)  =  (Homa `  C
)
31, 2arwhoma 15450 . 2  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) (Homa `  C
) (coda
`  F ) ) )
4 arwhom.j . . 3  |-  J  =  ( Hom  `  C
)
52, 4homahom 15444 . 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 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   2ndc2nd 6798   Hom chom 14722  domAcdoma 15425  codaccoda 15426  Natcarw 15427  Homachoma 15428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-1st 6799  df-2nd 6800  df-doma 15429  df-coda 15430  df-homa 15431  df-arw 15432
This theorem is referenced by: (None)
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