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Theorem arwhom 16902
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 2760 . . 3 (Homa𝐶) = (Homa𝐶)
31, 2arwhoma 16896 . 2 (𝐹𝐴𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)))
4 arwhom.j . . 3 𝐽 = (Hom ‘𝐶)
52, 4homahom 16890 . 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 1632  wcel 2139  cfv 6049  (class class class)co 6813  2nd c2nd 7332  Hom chom 16154  domacdoma 16871  codaccoda 16872  Arrowcarw 16873  Homachoma 16874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-1st 7333  df-2nd 7334  df-doma 16875  df-coda 16876  df-homa 16877  df-arw 16878
This theorem is referenced by: (None)
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