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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 29066 | . . . . . . 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 5948 | . . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺) | 
| 6 | 2, 3, 5 | 3eqtr4g 2802 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼) | 
| 7 | 6 | rexeqdv 3327 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒)) | 
| 8 | 4 | uhgrfun 29083 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) | 
| 9 | 8 | funfnd 6597 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) | 
| 10 | eleq2 2830 | . . . . . 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 1540 ∈ wcel 2108 ∃wrex 3070 dom cdm 5685 ran crn 5686 Fn wfn 6556 ‘cfv 6561 iEdgciedg 29014 Edgcedg 29064 UHGraphcuhgr 29073 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-edg 29065 df-uhgr 29075 | 
| This theorem is referenced by: vtxduhgr0edgnel 29512 | 
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