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Mirrors > Home > MPE Home > Th. List > uhgrn0 | Structured version Visualization version GIF version |
Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.) |
Ref | Expression |
---|---|
uhgrfun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uhgrn0 | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | uhgrfun.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 26847 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | fndm 6455 | . . . . . . 7 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴) | |
5 | 4 | feq2d 6500 | . . . . . 6 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
6 | 3, 5 | syl5ibcom 247 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
7 | 6 | imp 409 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
8 | 7 | ffvelrnda 6851 | . . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
9 | 8 | 3impa 1106 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
10 | eldifsni 4722 | . 2 ⊢ ((𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → (𝐸‘𝐹) ≠ ∅) | |
11 | 9, 10 | syl 17 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4567 dom cdm 5555 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 UHGraphcuhgr 26841 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-uhgr 26843 |
This theorem is referenced by: lpvtx 26853 subgruhgredgd 27066 |
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