uhgrn0 - Metamath Proof Explorer

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Theorem uhgrn0 29150

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 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		uhgrfun.e	. . . . . . 7 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	uhgrf 29145	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		fndm 6595	. . . . . . 7 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
5	4	feq2d 6646	. . . . . 6 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
6	3, 5	syl5ibcom 245	. . . . 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 7030	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
9	8	3impa 1110	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10		eldifsni 4734	. 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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∅c0 4274 𝒫 cpw 4542 {csn 4568 dom cdm 5624 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 Vtxcvtx 29079 iEdgciedg 29080 UHGraphcuhgr 29139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-uhgr 29141

This theorem is referenced by: lpvtx 29151 subgruhgredgd 29367

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