Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . 3 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
3 | 1, 2 | arwhoma 17293 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)(Homa‘𝐶)(coda‘𝐹))) |
4 | arwhom.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐶) | |
5 | 2, 4 | homahom 17287 | . 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 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 2nd c2nd 7677 Hom chom 16564 domacdoma 17268 codaccoda 17269 Arrowcarw 17270 Homachoma 17271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-1st 7678 df-2nd 7679 df-doma 17272 df-coda 17273 df-homa 17274 df-arw 17275 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |