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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 27112 | . . . . . . 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 5795 | . . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺) |
6 | 2, 3, 5 | 3eqtr4g 2799 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼) |
7 | 6 | rexeqdv 3319 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒)) |
8 | 4 | uhgrfun 27129 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
9 | 8 | funfnd 6400 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) |
10 | eleq2 2822 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖))) | |
11 | 10 | rexrn 6895 | . . . . 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 282 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
14 | 13 | adantr 484 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
15 | 14 | bicomd 226 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 dom cdm 5540 ran crn 5541 Fn wfn 6364 ‘cfv 6369 iEdgciedg 27060 Edgcedg 27110 UHGraphcuhgr 27119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-edg 27111 df-uhgr 27121 |
This theorem is referenced by: vtxduhgr0edgnel 27554 |
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