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Theorem nbuhgr2vtx1edgblem 26292
Description: Lemma for nbuhgr2vtx1edgb 26293. This reverse direction of nbgr2vtx1edg 26291 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 26278 . . 3 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
42eleq2i 2722 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
5 edguhgr 26069 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
64, 5sylan2b 491 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
71eqeq1i 2656 . . . . . . . . . . . . 13 (𝑉 = {𝑎, 𝑏} ↔ (Vtx‘𝐺) = {𝑎, 𝑏})
8 pweq 4194 . . . . . . . . . . . . . . 15 ((Vtx‘𝐺) = {𝑎, 𝑏} → 𝒫 (Vtx‘𝐺) = 𝒫 {𝑎, 𝑏})
98eleq2d 2716 . . . . . . . . . . . . . 14 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ∈ 𝒫 {𝑎, 𝑏}))
10 selpw 4198 . . . . . . . . . . . . . 14 (𝑒 ∈ 𝒫 {𝑎, 𝑏} ↔ 𝑒 ⊆ {𝑎, 𝑏})
119, 10syl6bb 276 . . . . . . . . . . . . 13 ((Vtx‘𝐺) = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
127, 11sylbi 207 . . . . . . . . . . . 12 (𝑉 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
1312adantl 481 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ {𝑎, 𝑏}))
14 prcom 4299 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
1514sseq1i 3662 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ 𝑒)
16 eqss 3651 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} = 𝑒 ↔ ({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}))
17 eleq1a 2725 . . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . 15 (({𝑎, 𝑏} ⊆ 𝑒𝑒 ⊆ {𝑎, 𝑏}) → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
2120ex 449 . . . . . . . . . . . . . 14 ({𝑎, 𝑏} ⊆ 𝑒 → (𝑒 ⊆ {𝑎, 𝑏} → (𝑒𝐸 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2215, 21sylbi 207 . . . . . . . . . . . . 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 763 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ⊆ {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2513, 24sylbid 230 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ 𝑒𝐸) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑏, 𝑎} ⊆ 𝑒 → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
2625ex 449 . . . . . . . . 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 3059 . . . . 5 (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → (∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒 → ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → {𝑎, 𝑏} ∈ 𝐸)))
31303impia 1280 . . . 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 232 . 2 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
34333impia 1280 1 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  wss 3607  𝒫 cpw 4191  {cpr 4212  cfv 5926  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  UHGraphcuhgr 25996   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-edg 25985  df-uhgr 25998  df-nbgr 26270
This theorem is referenced by:  nbuhgr2vtx1edgb  26293
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