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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 16914 | . . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻 |
4 | 3 | eleq2i 2831 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻) |
5 | 4 | biimpi 206 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻) |
6 | eqid 2760 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1 | arwrcl 16915 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
8 | 2, 6, 7 | homaf 16901 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
9 | ffn 6206 | . . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))) | |
10 | fnunirn 6675 | . . . . 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 222 | . . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)) |
13 | fveq2 6353 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | |
14 | df-ov 6817 | . . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | syl6eqr 2812 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) |
16 | 15 | eleq2d 2825 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦))) |
17 | 16 | rexxp 5420 | . . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
18 | 12, 17 | sylib 208 | . 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
19 | id 22 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦)) | |
20 | 2 | homadm 16911 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥) |
21 | 2 | homacd 16912 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦) |
22 | 20, 21 | oveq12d 6832 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦)) |
23 | 19, 22 | eleqtrrd 2842 | . . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
24 | 23 | rexlimivw 3167 | . . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
25 | 24 | rexlimivw 3167 | . 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 196 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 Vcvv 3340 𝒫 cpw 4302 〈cop 4327 ∪ cuni 4588 × cxp 5264 ran crn 5267 Fn wfn 6044 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 domacdoma 16891 codaccoda 16892 Arrowcarw 16893 Homachoma 16894 |
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 7115 |
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 6817 df-1st 7334 df-2nd 7335 df-doma 16895 df-coda 16896 df-homa 16897 df-arw 16898 |
This theorem is referenced by: arwdm 16918 arwcd 16919 arwhom 16922 arwdmcd 16923 coapm 16942 |
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