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Theorem nbuhgr2vtx1edgblem 27285
Description: Lemma for nbuhgr2vtx1edgb 27286. This reverse direction of nbgr2vtx1edg 27284 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 27274 . . 3 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
42eleq2i 2824 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
5 edguhgr 27066 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
64, 5sylan2b 597 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
71eqeq1i 2743 . . . . . . . . . . . . 13 (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8 pweq 4501 . . . . . . . . . . . . . . 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 4490 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
119, 10bitrdi 290 . . . . . . . . . . . . 13 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
127, 11sylbi 220 . . . . . . . . . . . 12 (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
1312adantl 485 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14 prcom 4620 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
1514sseq1i 3903 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16 eqss 3890 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}))
17 eleq1a 2828 . . . . . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . . . 15 (({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}) → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
2120ex 416 . . . . . . . . . . . . . 14 ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2215, 21sylbi 220 . . . . . . . . . . . . 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 727 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2513, 24sylbid 243 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2625ex 416 . . . . . . . . 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 455 . . . . . . 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 3192 . . . . 5 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
31303impia 1118 . . . 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 245 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
34333impia 1118 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wne 2934  wrex 3054  wss 3841  𝒫 cpw 4485  {cpr 4515  cfv 6333  (class class class)co 7164  Vtxcvtx 26933  Edgcedg 26984  UHGraphcuhgr 26993   NeighbVtx cnbgr 27266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-edg 26985  df-uhgr 26995  df-nbgr 27267
This theorem is referenced by:  nbuhgr2vtx1edgb  27286
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