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Theorem arwhom 17299
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 2818 . . 3 (Homa𝐶) = (Homa𝐶)
31, 2arwhoma 17293 . 2 (𝐹𝐴𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)))
4 arwhom.j . . 3 𝐽 = (Hom ‘𝐶)
52, 4homahom 17287 . 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 1528  wcel 2105  cfv 6348  (class class class)co 7145  2nd c2nd 7677  Hom chom 16564  domacdoma 17268  codaccoda 17269  Arrowcarw 17270  Homachoma 17271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-1st 7678  df-2nd 7679  df-doma 17272  df-coda 17273  df-homa 17274  df-arw 17275
This theorem is referenced by: (None)
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