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Theorem arwhom 17307
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 = (Arrow‘𝐶)
arwhom.j 𝐽 = (Hom ‘𝐶)
Assertion
Ref Expression
arwhom (𝐹𝐴 → (2nd𝐹) ∈ ((doma𝐹)𝐽(coda𝐹)))

Proof of Theorem arwhom
StepHypRef Expression
1 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
2 eqid 2801 . . 3 (Homa𝐶) = (Homa𝐶)
31, 2arwhoma 17301 . 2 (𝐹𝐴𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)))
4 arwhom.j . . 3 𝐽 = (Hom ‘𝐶)
52, 4homahom 17295 . 2 (𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)) → (2nd𝐹) ∈ ((doma𝐹)𝐽(coda𝐹)))
63, 5syl 17 1 (𝐹𝐴 → (2nd𝐹) ∈ ((doma𝐹)𝐽(coda𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  2nd c2nd 7674  Hom chom 16572  domacdoma 17276  codaccoda 17277  Arrowcarw 17278  Homachoma 17279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-1st 7675  df-2nd 7676  df-doma 17280  df-coda 17281  df-homa 17282  df-arw 17283
This theorem is referenced by: (None)
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