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Theorem nbuhgr2vtx1edgblem 28473
Description: Lemma for nbuhgr2vtx1edgb 28474. This reverse direction of nbgr2vtx1edg 28472 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.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbgr2vtx1edg.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrel 28462 . . 3 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
42eleq2i 2824 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
5 edguhgr 28254 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
64, 5sylan2b 594 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
71eqeq1i 2736 . . . . . . . . . . . . 13 (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8 pweq 4610 . . . . . . . . . . . . . . 15 ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
98eleq2d 2818 . . . . . . . . . . . . . 14 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10 velpw 4601 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
119, 10bitrdi 286 . . . . . . . . . . . . 13 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
127, 11sylbi 216 . . . . . . . . . . . 12 (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
1312adantl 482 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14 prcom 4729 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
1514sseq1i 4006 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16 eqss 3993 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}))
17 eleq1a 2827 . . . . . . . . . . . . . . . . . 18 (𝑒𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸))
1817a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (𝑒𝐸 → ({𝑎, 𝑏} = 𝑒 → {𝑎, 𝑏} ∈ 𝐸)))
1918com13 88 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
2016, 19sylbir 234 . . . . . . . . . . . . . . 15 (({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}) → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
2120ex 413 . . . . . . . . . . . . . 14 ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2215, 21sylbi 216 . . . . . . . . . . . . 13 ({𝑏, 𝑎} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2322com13 88 . . . . . . . . . . . 12 (𝑒𝐸 → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2423ad2antlr 725 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2513, 24sylbid 239 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2625ex 413 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸)))))
276, 26mpid 44 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → (𝑉 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2827impancom 452 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2928com14 96 . . . . . 6 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (𝑒𝐸 → ({𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))))
3029rexlimdv 3152 . . . . 5 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
31303impia 1117 . . . 4 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒) → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸))
3231com12 32 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒) → {𝑎, 𝑏} ∈ 𝐸))
333, 32biimtrid 241 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
34333impia 1117 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wrex 3069  wss 3944  𝒫 cpw 4596  {cpr 4624  cfv 6532  (class class class)co 7393  Vtxcvtx 28121  Edgcedg 28172  UHGraphcuhgr 28181   NeighbVtx cnbgr 28454
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 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
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 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-edg 28173  df-uhgr 28183  df-nbgr 28455
This theorem is referenced by:  nbuhgr2vtx1edgb  28474
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