Theorem nbuhgr2vtx1edgblem 28362

Description: Lemma for nbuhgr2vtx1edgb 28363. This reverse direction of nbgr2vtx1edg 28361 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 28351	. . 3 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))
4	2	eleq2i 2824	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
5		edguhgr 28143	. . . . . . . . . 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 4579	. . . . . . . . . . . . . . 15 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
9	8	eleq2d 2818	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10		velpw 4570	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
11	9, 10	bitrdi 286	. . . . . . . . . . . . 13 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
12	7, 11	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
13	12	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14		prcom 4698	. . . . . . . . . . . . . . 15 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
15	14	sseq1i 3975	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16		eqss 3962	. . . . . . . . . . . . . . . 16 ⊢ ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}))
17		eleq1a 2827	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 15 ⊢ (({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
21	20	ex 413	. . . . . . . . . . . . . 14 ⊢ ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
22	15, 21	sylbi 216	. . . . . . . . . . . . 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 725	. . . . . . . . . . 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 239	. . . . . . . . . 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 413	. . . . . . . . 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 452	. . . . . . 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 3146	. . . . 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 241	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069   ⊆ wss 3913  𝒫 cpw 4565  {cpr 4593  ‘cfv 6501  (class class class)co 7362  Vtxcvtx 28010  Edgcedg 28061  UHGraphcuhgr 28070   NeighbVtx cnbgr 28343

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-edg 28062  df-uhgr 28072  df-nbgr 28344

This theorem is referenced by:  nbuhgr2vtx1edgb  28363