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 2736 | . . 3 ⊢ (Homa‘𝐶) = (Homa‘𝐶) | |
3 | 1, 2 | arwhoma 17505 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)(Homa‘𝐶)(coda‘𝐹))) |
4 | arwhom.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐶) | |
5 | 2, 4 | homahom 17499 | . 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 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 2nd c2nd 7738 Hom chom 16760 domacdoma 17480 codaccoda 17481 Arrowcarw 17482 Homachoma 17483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-1st 7739 df-2nd 7740 df-doma 17484 df-coda 17485 df-homa 17486 df-arw 17487 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |