Theorem nbuhgr2vtx1edgblem 29386

Description: Lemma for nbuhgr2vtx1edgb 29387. This reverse direction of nbgr2vtx1edg 29385 only holds for classes whose edges are subsets of the set of vertices, which is the property of hypergraphs. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)

Hypotheses

Ref	Expression
nbgr2vtx1edg.v	⊢ 𝑉 = (Vtx‘𝐺)
nbgr2vtx1edg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
nbuhgr2vtx1edgblem	⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)

Distinct variable groups:   𝐸,𝑎,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎,𝑏

Proof of Theorem nbuhgr2vtx1edgblem

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgr2vtx1edg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		nbgr2vtx1edg.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	nbgrel 29375	. . 3 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
4	2	eleq2i 2836	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
5		edguhgr 29164	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
6	4, 5	sylan2b 593	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
7	1	eqeq1i 2745	. . . . . . . . . . . . 13 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8		pweq 4636	. . . . . . . . . . . . . . 15 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
9	8	eleq2d 2830	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10		velpw 4627	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
11	9, 10	bitrdi 287	. . . . . . . . . . . . 13 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
12	7, 11	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
13	12	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14		prcom 4757	. . . . . . . . . . . . . . 15 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
15	14	sseq1i 4037	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16		eqss 4024	. . . . . . . . . . . . . . . 16 ⊢ ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}))
17		eleq1a 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ 𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸))
18	17	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (𝑒 ∈ 𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸)))
19	18	com13 88	. . . . . . . . . . . . . . . 16 ⊢ ({𝑎, 𝑏} = 𝑒 → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
20	16, 19	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ (({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
21	20	ex 412	. . . . . . . . . . . . . 14 ⊢ ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
22	15, 21	sylbi 217	. . . . . . . . . . . . 13 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
23	22	com13 88	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
24	23	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
25	13, 24	sylbid 240	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
26	25	ex 412	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))))
27	6, 26	mpid 44	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑉 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
28	27	impancom 451	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
29	28	com14 96	. . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (𝑒 ∈ 𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))))
30	29	rexlimdv 3159	. . . . 5 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
31	30	3impia 1117	. . . 4 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))
32	31	com12 32	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) → {𝑎, 𝑏} ∈ 𝐸))
33	3, 32	biimtrid 242	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
34	33	3impia 1117	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ⊆ wss 3976  𝒫 cpw 4622  {cpr 4650  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082  UHGraphcuhgr 29091   NeighbVtx cnbgr 29367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-edg 29083  df-uhgr 29093  df-nbgr 29368

This theorem is referenced by:  nbuhgr2vtx1edgb  29387