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Mirrors > Home > MPE Home > Th. List > arwhoma | Structured version Visualization version GIF version |
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwhoma.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
arwhoma | ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwrcl.a | . . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶) | |
2 | arwhoma.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | 1, 2 | arwval 17302 | . . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻 |
4 | 3 | eleq2i 2904 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻) |
5 | 4 | biimpi 218 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻) |
6 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1 | arwrcl 17303 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
8 | 2, 6, 7 | homaf 17289 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
9 | ffn 6513 | . . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))) | |
10 | fnunirn 7011 | . . . . 5 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) |
12 | 5, 11 | mpbid 234 | . . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)) |
13 | fveq2 6669 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | |
14 | df-ov 7158 | . . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | syl6eqr 2874 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) |
16 | 15 | eleq2d 2898 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦))) |
17 | 16 | rexxp 5712 | . . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
18 | 12, 17 | sylib 220 | . 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
19 | id 22 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦)) | |
20 | 2 | homadm 17299 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥) |
21 | 2 | homacd 17300 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦) |
22 | 20, 21 | oveq12d 7173 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦)) |
23 | 19, 22 | eleqtrrd 2916 | . . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
24 | 23 | rexlimivw 3282 | . . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
25 | 24 | rexlimivw 3282 | . 2 ⊢ (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
26 | 18, 25 | syl 17 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 𝒫 cpw 4538 〈cop 4572 ∪ cuni 4837 × cxp 5552 ran crn 5555 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 domacdoma 17279 codaccoda 17280 Arrowcarw 17281 Homachoma 17282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-1st 7688 df-2nd 7689 df-doma 17283 df-coda 17284 df-homa 17285 df-arw 17286 |
This theorem is referenced by: arwdm 17306 arwcd 17307 arwhom 17310 arwdmcd 17311 coapm 17330 |
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