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Theorem uhgrn0 25962

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 2622	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 25957	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		fndm 5990	. . . . . . 7 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6031	. . . . . 6 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
6	3, 5	syl5ibcom 235	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
7	6	imp 445	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
8	7	ffvelrnda 6359	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
9	8	3impa 1259	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10		eldifsni 4320	. 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {csn 4177 dom cdm 5114 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-uhgr 25953

This theorem is referenced by: lpvtx 25963 subgruhgredgd 26176

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