arwhom - Metamath Proof Explorer

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Theorem arwhom 18173

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 18167	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((dom_a‘𝐹)(Hom_a‘𝐶)(cod_a‘𝐹)))
4		arwhom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
5	2, 4	homahom 18161	. 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 1570 ∈ wcel 2145 ‘cfv 6528 (class class class)co 7409 2^nd c2nd 7984 Hom chom 17386 dom_acdoma 18142 cod_accoda 18143 Arrowcarw 18144 Hom_achoma 18145

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5543 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-ov 7412 df-1st 7985 df-2nd 7986 df-doma 18146 df-coda 18147 df-homa 18148 df-arw 18149

This theorem is used by: termcarweu 50552

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