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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 26836 | . . . . . . 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 5809 | . . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺) |
6 | 2, 3, 5 | 3eqtr4g 2883 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼) |
7 | 6 | rexeqdv 3418 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒)) |
8 | 4 | uhgrfun 26853 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
9 | 8 | funfnd 6388 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) |
10 | eleq2 2903 | . . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖))) | |
11 | 10 | rexrn 6855 | . . . . 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 281 | . . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
14 | 13 | adantr 483 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖))) |
15 | 14 | bicomd 225 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 dom cdm 5557 ran crn 5558 Fn wfn 6352 ‘cfv 6357 iEdgciedg 26784 Edgcedg 26834 UHGraphcuhgr 26843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-edg 26835 df-uhgr 26845 |
This theorem is referenced by: vtxduhgr0edgnel 27278 |
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