uhgrn0 - Metamath Proof Explorer

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Theorem uhgrn0 28893

Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)

**Hypothesis**
Ref	Expression
uhgrfun.e	⊢ 𝐸 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
uhgrn0	⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)

**Proof of Theorem uhgrn0**
Step	Hyp	Ref	Expression
1		eqid 2728	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 28888	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		fndm 6657	. . . . . . 7 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6708	. . . . . 6 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
6	3, 5	syl5ibcom 244	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
7	6	imp 406	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
8	7	ffvelcdmda 7094	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
9	8	3impa 1108	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10		eldifsni 4794	. 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ∅c0 4323 𝒫 cpw 4603 {csn 4629 dom cdm 5678 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 Vtxcvtx 28822 iEdgciedg 28823 UHGraphcuhgr 28882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-uhgr 28884

This theorem is referenced by: lpvtx 28894 subgruhgredgd 29110

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