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Theorem nbgr2vtx1edg 27059
Description: If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Revised by AV, 25-Mar-2021.) (Proof shortened by AV, 13-Feb-2022.)
Hypotheses
Ref Expression
nbgr2vtx1edg.v 𝑉 = (Vtx‘𝐺)
nbgr2vtx1edg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbgr2vtx1edg (((♯‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))
Distinct variable groups:   𝑛,𝐸   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣
Allowed substitution hint:   𝐸(𝑣)

Proof of Theorem nbgr2vtx1edg
Dummy variables 𝑎 𝑏 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5 𝑉 = (Vtx‘𝐺)
21fvexi 6677 . . . 4 𝑉 ∈ V
3 hash2prb 13818 . . . 4 (𝑉 ∈ V → ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
42, 3ax-mp 5 . . 3 ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}))
5 simpll 763 . . . . . . . . . 10 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎𝑉𝑏𝑉))
65ancomd 462 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏𝑉𝑎𝑉))
7 simpl 483 . . . . . . . . . . 11 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑎𝑏)
87necomd 3068 . . . . . . . . . 10 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑏𝑎)
98ad2antlr 723 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏𝑎)
10 id 22 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
11 sseq2 3990 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏}))
1211adantl 482 . . . . . . . . . . 11 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏}))
13 ssidd 3987 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑎, 𝑏})
1410, 12, 13rspcedvd 3623 . . . . . . . . . 10 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
1514adantl 482 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
16 nbgr2vtx1edg.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
171, 16nbgrel 27049 . . . . . . . . 9 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
186, 9, 15, 17syl3anbrc 1335 . . . . . . . 8 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
197ad2antlr 723 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎𝑏)
20 sseq2 3990 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
2120adantl 482 . . . . . . . . . . 11 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
22 prcom 4660 . . . . . . . . . . . . 13 {𝑏, 𝑎} = {𝑎, 𝑏}
2322eqimssi 4022 . . . . . . . . . . . 12 {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
2423a1i 11 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
2510, 21, 24rspcedvd 3623 . . . . . . . . . 10 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
2625adantl 482 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
271, 16nbgrel 27049 . . . . . . . . 9 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
285, 19, 26, 27syl3anbrc 1335 . . . . . . . 8 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
29 difprsn1 4725 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
3029raleqdv 3413 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
31 vex 3495 . . . . . . . . . . . . 13 𝑏 ∈ V
32 eleq1 2897 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
3331, 32ralsn 4611 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
3430, 33syl6bb 288 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
35 difprsn2 4726 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
3635raleqdv 3413 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
37 vex 3495 . . . . . . . . . . . . 13 𝑎 ∈ V
38 eleq1 2897 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
3937, 38ralsn 4611 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
4036, 39syl6bb 288 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
4134, 40anbi12d 630 . . . . . . . . . 10 (𝑎𝑏 → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
4241adantr 481 . . . . . . . . 9 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
4342ad2antlr 723 . . . . . . . 8 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
4418, 28, 43mpbir2and 709 . . . . . . 7 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
4544ex 413 . . . . . 6 (((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))
46 eleq1 2897 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47 id 22 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48 difeq1 4089 . . . . . . . . . . . 12 (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
4948raleqdv 3413 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5047, 49raleqbidv 3399 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51 sneq 4567 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → {𝑣} = {𝑎})
5251difeq2d 4096 . . . . . . . . . . . 12 (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
53 oveq2 7153 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
5453eleq2d 2895 . . . . . . . . . . . 12 (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
5552, 54raleqbidv 3399 . . . . . . . . . . 11 (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
56 sneq 4567 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑣} = {𝑏})
5756difeq2d 4096 . . . . . . . . . . . 12 (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
58 oveq2 7153 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
5958eleq2d 2895 . . . . . . . . . . . 12 (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6057, 59raleqbidv 3399 . . . . . . . . . . 11 (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6137, 31, 55, 60ralpr 4628 . . . . . . . . . 10 (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6250, 61syl6bb 288 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))
6346, 62imbi12d 346 . . . . . . . 8 (𝑉 = {𝑎, 𝑏} → ((𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
6463adantl 482 . . . . . . 7 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → ((𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
6564adantl 482 . . . . . 6 (((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ((𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))))
6645, 65mpbird 258 . . . . 5 (((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
6766ex 413 . . . 4 ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
6867rexlimivv 3289 . . 3 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
694, 68sylbi 218 . 2 ((♯‘𝑉) = 2 → (𝑉𝐸 → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
7069imp 407 1 (((♯‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  2c2 11680  chash 13678  Vtxcvtx 26708  Edgcedg 26759   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-nbgr 27042
This theorem is referenced by:  uvtx2vtx1edg  27107
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