Theorem arwhom 17766

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	⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹)))

Proof of Theorem arwhom

Step	Hyp	Ref	Expression
1		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
2		eqid 2738	. . 3 ⊢ (Homa‘𝐶) = (Homa‘𝐶)
3	1, 2	arwhoma 17760	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)(Homa‘𝐶)(coda‘𝐹)))
4		arwhom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
5	2, 4	homahom 17754	. 2 ⊢ (𝐹 ∈ ((doma‘𝐹)(Homa‘𝐶)(coda‘𝐹)) → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹)))
6	3, 5	syl 17	1 ⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  2^nd c2nd 7830  Hom chom 16973  dom_acdoma 17735  cod_accoda 17736  Arrowcarw 17737  Hom_achoma 17738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-1st 7831  df-2nd 7832  df-doma 17739  df-coda 17740  df-homa 17741  df-arw 17742

This theorem is referenced by: (None)