arwhom - Metamath Proof Explorer

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Theorem arwhom 18252

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 2763	. . 3 ⊢ (Hom_a‘𝐶) = (Hom_a‘𝐶)
3	1, 2	arwhoma 18246	. 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((dom_a‘𝐹)(Hom_a‘𝐶)(cod_a‘𝐹)))
4		arwhom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
5	2, 4	homahom 18240	. 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 2147 ‘cfv 6555 (class class class)co 7440 2^nd c2nd 8021 Hom chom 17465 dom_acdoma 18221 cod_accoda 18222 Arrowcarw 18223 Hom_achoma 18224

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-1st 8022 df-2nd 8023 df-doma 18225 df-coda 18226 df-homa 18227 df-arw 18228

This theorem is used by: termcarweu 50866

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