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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.) (Revised by AV, 6-Jul-2022.) |
Ref | Expression |
---|---|
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 | edgval 29081 | . . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
3 | uhgrvtxedgiedgb.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | uhgrvtxedgiedgb.i | . . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | rneqi 5951 | . . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺) |
6 | 2, 3, 5 | 3eqtr4g 2800 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼) |
7 | 6 | rexeqdv 3325 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒)) |
8 | 4 | uhgrfun 29098 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
9 | 8 | funfnd 6599 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) |
10 | eleq2 2828 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖))) | |
11 | 10 | rexrn 7107 | . . . . 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 | 7, 12 | bitrd 279 | . . 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 223 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 dom cdm 5689 ran crn 5690 Fn wfn 6558 ‘cfv 6563 iEdgciedg 29029 Edgcedg 29079 UHGraphcuhgr 29088 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-edg 29080 df-uhgr 29090 |
This theorem is referenced by: vtxduhgr0edgnel 29527 |
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