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Theorem arwhom 13899
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 2296 . . 3  |-  (Homa `  C
)  =  (Homa `  C
)
31, 2arwhoma 13893 . 2  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) (Homa `  C
) (coda
`  F ) ) )
4 arwhom.j . . 3  |-  J  =  (  Hom  `  C
)
52, 4homahom 13887 . 2  |-  ( F  e.  ( (domA `  F ) (Homa
`  C ) (coda `  F ) )  -> 
( 2nd `  F
)  e.  ( (domA `  F
) J (coda `  F
) ) )
63, 5syl 15 1  |-  ( F  e.  A  ->  ( 2nd `  F )  e.  ( (domA `  F ) J (coda `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137    Hom chom 13235  domAcdoma 13868  codaccoda 13869  Natcarw 13870  Homachoma 13871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875
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