arwhom - Metamath Proof Explorer

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Theorem arwhom 18226

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 → (2^nd ‘𝐹) ∈ ((dom_a‘𝐹)𝐽(cod_a‘𝐹)))

**Proof of Theorem arwhom**
Step	Hyp	Ref	Expression
1		arwrcl.a	. . 3 ⊢ 𝐴 = (Arrow‘𝐶)
2		eqid 2760	. . 3 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
3	1, 2	arwhoma 18220	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((dom_a‘𝐹)(Hom_a‘𝐶)(cod_a‘𝐹)))
4		arwhom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
5	2, 4	homahom 18214	. 2 ⊢ (𝐹 ∈ ((dom_a‘𝐹)(Hom_a‘𝐶)(cod_a‘𝐹)) → (2^nd ‘𝐹) ∈ ((dom_a‘𝐹)𝐽(cod_a‘𝐹)))
6	3, 5	syl 18	1 ⊢ (𝐹 ∈ 𝐴 → (2^nd ‘𝐹) ∈ ((dom_a‘𝐹)𝐽(cod_a‘𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ‘cfv 6538 (class class class)co 7420 2^nd c2nd 8000 Hom chom 17439 dom_acdoma 18195 cod_accoda 18196 Arrowcarw 18197 Hom_achoma 18198

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-1st 8001 df-2nd 8002 df-doma 18199 df-coda 18200 df-homa 18201 df-arw 18202

This theorem is used by: termcarweu 50636

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