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Mirrors > Home > MPE Home > Th. List > uhgrvtxedgiedgb | Structured version Visualization version GIF version |
Description: In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) |
Ref | Expression |
---|---|
uhgrvtxedgiedgb.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrvtxedgiedgb.i | ⊢ 𝐼 = (iEdg‘𝐺) |
uhgrvtxedgiedgb.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uhgrvtxedgiedgb | ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖) ↔ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgaval 25794 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) | |
2 | uhgrvtxedgiedgb.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | uhgrvtxedgiedgb.i | . . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | rneqi 5273 | . . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺) |
5 | 1, 2, 4 | 3eqtr4g 2669 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼) |
6 | 5 | rexeqdv 3122 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒)) |
7 | 3 | uhgrfun 25732 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
8 | funfn 5833 | . . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼) | |
9 | 7, 8 | sylib 207 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) |
10 | eleq2 2677 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖))) | |
11 | 10 | rexrn 6269 | . . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
13 | 6, 12 | bitrd 267 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
14 | 13 | adantr 480 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
15 | 14 | bicomd 212 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖) ↔ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 Edgcedga 25792 |
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-sep 4709 ax-nul 4717 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-ral 2901 df-rex 2902 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-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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-uhgr 25724 df-edga 25793 |
This theorem is referenced by: vtxduhgr0edgnel 40709 |
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