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Theorem nbuhgr2vtx1edgblem 27718
Description: Lemma for nbuhgr2vtx1edgb 27719. This reverse direction of nbgr2vtx1edg 27717 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 27707 . . 3 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
42eleq2i 2830 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
5 edguhgr 27499 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
64, 5sylan2b 594 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
71eqeq1i 2743 . . . . . . . . . . . . 13 (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8 pweq 4549 . . . . . . . . . . . . . . 15 ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
98eleq2d 2824 . . . . . . . . . . . . . 14 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10 velpw 4538 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
119, 10bitrdi 287 . . . . . . . . . . . . 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 4668 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
1514sseq1i 3949 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16 eqss 3936 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}))
17 eleq1a 2834 . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . 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 3212 . . . . 5 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
31303impia 1116 . . . 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, 32syl5bi 241 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
34333impia 1116 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 1086   = wceq 1539  wcel 2106  wne 2943  wrex 3065  wss 3887  𝒫 cpw 4533  {cpr 4563  cfv 6433  (class class class)co 7275  Vtxcvtx 27366  Edgcedg 27417  UHGraphcuhgr 27426   NeighbVtx cnbgr 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-edg 27418  df-uhgr 27428  df-nbgr 27700
This theorem is referenced by:  nbuhgr2vtx1edgb  27719
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