Theorem nbuhgr2vtx1edgblem 29329

Description: Lemma for nbuhgr2vtx1edgb 29330. This reverse direction of nbgr2vtx1edg 29328 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 29318	. . 3 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
4	2	eleq2i 2823	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
5		edguhgr 29107	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
6	4, 5	sylan2b 594	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
7	1	eqeq1i 2736	. . . . . . . . . . . . 13 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8		pweq 4561	. . . . . . . . . . . . . . 15 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
9	8	eleq2d 2817	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10		velpw 4552	. . . . . . . . . . . . . 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 4682	. . . . . . . . . . . . . . 15 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
15	14	sseq1i 3958	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16		eqss 3945	. . . . . . . . . . . . . . . 16 ⊢ ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}))
17		eleq1a 2826	. . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . 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 3131	. . . . 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 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4547  {cpr 4575  ‘cfv 6481  (class class class)co 7346  Vtxcvtx 28974  Edgcedg 29025  UHGraphcuhgr 29034   NeighbVtx cnbgr 29310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-edg 29026  df-uhgr 29036  df-nbgr 29311

This theorem is referenced by:  nbuhgr2vtx1edgb  29330