Theorem nbuhgr2vtx1edgblem 40573

Description: Lemma for nbuhgr2vtx1edgb 40574. This reverse direction of nbgr2vtx1edg 40572 only holds for classes whose edges are subsets of the set of vertices (hypergraphs!) (Contributed by AV, 2-Nov-2020.)

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	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		nbgr2vtx1edg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	nbgrel 40564	. . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
4	3	adantr 480	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
5	2	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
6		edguhgr 25803	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
7	5, 6	sylan2b 491	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
8	1	eqeq1i 2615	. . . . . . . . . . . . 13 ⊢ (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
9		pweq 4111	. . . . . . . . . . . . . . 15 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
10	9	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
11		selpw 4115	. . . . . . . . . . . . . 14 ⊢ (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
12	10, 11	syl6bb 275	. . . . . . . . . . . . 13 ⊢ ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
13	8, 12	sylbi 206	. . . . . . . . . . . 12 ⊢ (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14	13	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
15		prcom 4211	. . . . . . . . . . . . . . . 16 ⊢ {𝑏, 𝑎} = {𝑎, 𝑏}
16	15	sseq1i 3592	. . . . . . . . . . . . . . 15 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
17		eqss 3583	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}))
18		eleq1a 2683	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ 𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸))
19	18	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (𝑒 ∈ 𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸)))
20	19	com13 86	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑎, 𝑏} = 𝑒 → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
21	17, 20	sylbir 224	. . . . . . . . . . . . . . . 16 ⊢ (({𝑎, 𝑏} ⊆ 𝑒 ∧ 𝑒 ⊆ {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
22	21	ex 449	. . . . . . . . . . . . . . 15 ⊢ ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
23	16, 22	sylbi 206	. . . . . . . . . . . . . 14 ⊢ ({𝑏, 𝑎} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
24	23	com13 86	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ 𝐸 → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
26	25	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
27	14, 26	sylbid 229	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
28	27	ex 449	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))))
29	7, 28	mpid 43	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ 𝐸) → (𝑉 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
30	29	impancom 455	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
31	30	com14 94	. . . . . 6 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (𝑒 ∈ 𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))))
32	31	rexlimdv 3012	. . . . 5 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
33	32	3impia 1253	. . . 4 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))
34	33	com12 32	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) → {𝑎, 𝑏} ∈ 𝐸))
35	4, 34	sylbid 229	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
36	35	3impia 1253	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   UHGraph cuhgr 25722  Edgcedga 25792   NeighbVtx cnbgr 40550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-uhgr 25724  df-edga 25793  df-nbgr 40554

This theorem is referenced by:  nbuhgr2vtx1edgb  40574