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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 16516 | . . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻 |
4 | 3 | eleq2i 2680 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻) |
5 | 4 | biimpi 205 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻) |
6 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1 | arwrcl 16517 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
8 | 2, 6, 7 | homaf 16503 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
9 | ffn 5958 | . . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))) | |
10 | fnunirn 6415 | . . . . 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 221 | . . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)) |
13 | fveq2 6103 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | |
14 | df-ov 6552 | . . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | syl6eqr 2662 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) |
16 | 15 | eleq2d 2673 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦))) |
17 | 16 | rexxp 5186 | . . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
18 | 12, 17 | sylib 207 | . 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
19 | id 22 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦)) | |
20 | 2 | homadm 16513 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥) |
21 | 2 | homacd 16514 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦) |
22 | 20, 21 | oveq12d 6567 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦)) |
23 | 19, 22 | eleqtrrd 2691 | . . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
24 | 23 | rexlimivw 3011 | . . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
25 | 24 | rexlimivw 3011 | . 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 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 𝒫 cpw 4108 〈cop 4131 ∪ cuni 4372 × cxp 5036 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 domacdoma 16493 codaccoda 16494 Arrowcarw 16495 Homachoma 16496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-1st 7059 df-2nd 7060 df-doma 16497 df-coda 16498 df-homa 16499 df-arw 16500 |
This theorem is referenced by: arwdm 16520 arwcd 16521 arwhom 16524 arwdmcd 16525 coapm 16544 |
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