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Theorem nbuhgr2vtx1edgblem 28608
Description: Lemma for nbuhgr2vtx1edgb 28609. This reverse direction of nbgr2vtx1edg 28607 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 28597 . . 3 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
42eleq2i 2826 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
5 edguhgr 28389 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
64, 5sylan2b 595 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
71eqeq1i 2738 . . . . . . . . . . . . 13 (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8 pweq 4617 . . . . . . . . . . . . . . 15 ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
98eleq2d 2820 . . . . . . . . . . . . . 14 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10 velpw 4608 . . . . . . . . . . . . . 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 483 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14 prcom 4737 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
1514sseq1i 4011 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16 eqss 3998 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}))
17 eleq1a 2829 . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 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 726 . . . . . . . . . . 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 414 . . . . . . . . 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 453 . . . . . . 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 3154 . . . . 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, 32biimtrid 241 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wrex 3071  wss 3949  𝒫 cpw 4603  {cpr 4631  cfv 6544  (class class class)co 7409  Vtxcvtx 28256  Edgcedg 28307  UHGraphcuhgr 28316   NeighbVtx cnbgr 28589
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-edg 28308  df-uhgr 28318  df-nbgr 28590
This theorem is referenced by:  nbuhgr2vtx1edgb  28609
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