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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 2739 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | uhgrfun.e | . . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 27019 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | fndm 6450 | . . . . . . 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 248 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
7 | 6 | imp 410 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
8 | 7 | ffvelrnda 6873 | . . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
9 | 8 | 3impa 1111 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅})) |
10 | eldifsni 4688 | . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∖ cdif 3850 ∅c0 4221 𝒫 cpw 4498 {csn 4526 dom cdm 5535 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 Vtxcvtx 26953 iEdgciedg 26954 UHGraphcuhgr 27013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-uhgr 27015 |
This theorem is referenced by: lpvtx 27025 subgruhgredgd 27238 |
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