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Mirrors > Home > MPE Home > Th. List > arwhom | Structured version Visualization version GIF version |
Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwhom.j | ⊢ 𝐽 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
arwhom | ⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwrcl.a | . . 3 ⊢ 𝐴 = (Arrow‘𝐶) | |
2 | eqid 2760 | . . 3 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
3 | 1, 2 | arwhoma 16896 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)(Homa‘𝐶)(coda‘𝐹))) |
4 | arwhom.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐶) | |
5 | 2, 4 | homahom 16890 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6813 2nd c2nd 7332 Hom chom 16154 domacdoma 16871 codaccoda 16872 Arrowcarw 16873 Homachoma 16874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-1st 7333 df-2nd 7334 df-doma 16875 df-coda 16876 df-homa 16877 df-arw 16878 |
This theorem is referenced by: (None) |
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