Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrncvvdeqlem4 Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlem4 41472
 Description: Lemma 4 for frgrncvvdeq 41480. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem4 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦   𝑥,𝑦   𝑦,𝐸   𝑦,𝑁
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐷(𝑥)   𝐸(𝑥)   𝐺(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem4
Dummy variables 𝑛 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
21ineq2i 3773 . 2 ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))
3 frgrncvvdeq.nx . . . . . . 7 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2680 . . . . . 6 (𝑥𝐷𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5 frgrncvvdeq.f . . . . . . . 8 (𝜑𝐺 ∈ FriendGraph )
6 frgrusgr 41432 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
75, 6syl 17 . . . . . . 7 (𝜑𝐺 ∈ USGraph )
8 frgrncvvdeq.v1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
98nbgrisvtx 40581 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑥 ∈ (𝐺 NeighbVtx 𝑋)) → 𝑥𝑉)
109ex 449 . . . . . . 7 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
117, 10syl 17 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥𝑉))
124, 11syl5bi 231 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
1312imp 444 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
14 frgrncvvdeq.e . . . . 5 𝐸 = (Edg‘𝐺)
15 frgrncvvdeq.x . . . . 5 (𝜑𝑋𝑉)
16 frgrncvvdeq.y . . . . 5 (𝜑𝑌𝑉)
17 frgrncvvdeq.ne . . . . 5 (𝜑𝑋𝑌)
18 frgrncvvdeq.xy . . . . 5 (𝜑𝑌𝐷)
19 frgrncvvdeq.a . . . . 5 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
208, 14, 3, 1, 15, 16, 17, 18, 5, 19frgrncvvdeqlem1 41469 . . . 4 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
215adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝐺 ∈ FriendGraph )
228, 14frcond3 41440 . . . . 5 (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛})
2321, 22syl 17 . . . 4 ((𝜑𝑥𝐷) → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛})
2413, 20, 23jca31 555 . . 3 ((𝜑𝑥𝐷) → ((𝑥𝑉𝑌 ∈ (𝑉 ∖ {𝑥})) ∧ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}))
25 sneq 4135 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑎} = {𝑥})
2625difeq2d 3690 . . . . . . . 8 (𝑎 = 𝑥 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑥}))
27 oveq2 6557 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑥))
2827ineq1d 3775 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)))
2928eqeq1d 2612 . . . . . . . . 9 (𝑎 = 𝑥 → (((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} ↔ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}))
3029rexbidv 3034 . . . . . . . 8 (𝑎 = 𝑥 → (∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} ↔ ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}))
3126, 30raleqbidv 3129 . . . . . . 7 (𝑎 = 𝑥 → (∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} ↔ ∀𝑏 ∈ (𝑉 ∖ {𝑥})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}))
3231rspcva 3280 . . . . . 6 ((𝑥𝑉 ∧ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}) → ∀𝑏 ∈ (𝑉 ∖ {𝑥})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛})
33 oveq2 6557 . . . . . . . . . . . 12 (𝑏 = 𝑌 → (𝐺 NeighbVtx 𝑏) = (𝐺 NeighbVtx 𝑌))
3433ineq2d 3776 . . . . . . . . . . 11 (𝑏 = 𝑌 → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)))
3534eqeq1d 2612 . . . . . . . . . 10 (𝑏 = 𝑌 → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} ↔ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
3635rexbidv 3034 . . . . . . . . 9 (𝑏 = 𝑌 → (∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} ↔ ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
3736rspcva 3280 . . . . . . . 8 ((𝑌 ∈ (𝑉 ∖ {𝑥}) ∧ ∀𝑏 ∈ (𝑉 ∖ {𝑥})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}) → ∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛})
38 vsnid 4156 . . . . . . . . . . . . . . 15 𝑛 ∈ {𝑛}
39 eleq2 2677 . . . . . . . . . . . . . . . . 17 ({𝑛} = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) → (𝑛 ∈ {𝑛} ↔ 𝑛 ∈ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))))
4039eqcoms 2618 . . . . . . . . . . . . . . . 16 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ {𝑛} ↔ 𝑛 ∈ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))))
41 elin 3758 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
4241biimpi 205 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
4340, 42syl6bi 242 . . . . . . . . . . . . . . 15 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌))))
4438, 43mpi 20 . . . . . . . . . . . . . 14 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
4514nbusgreledg 40575 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
46 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑛, 𝑥} = {𝑥, 𝑛}
4746eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
4845, 47syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
4948biimpd 218 . . . . . . . . . . . . . . . . . . . 20 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
507, 49syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
5150adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐷) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
5251com12 32 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
5352adantr 480 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → ((𝜑𝑥𝐷) → {𝑥, 𝑛} ∈ 𝐸))
5453imp 444 . . . . . . . . . . . . . . 15 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
551eqcomi 2619 . . . . . . . . . . . . . . . . . . 19 (𝐺 NeighbVtx 𝑌) = 𝑁
5655eleq2i 2680 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛𝑁)
5756biimpi 205 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑁)
5857adantl 481 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛𝑁)
598, 14, 3, 1, 15, 16, 17, 18, 5, 19frgrncvvdeqlem3 41471 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)
60 preq2 4213 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
6160eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
6261riota2 6533 . . . . . . . . . . . . . . . 16 ((𝑛𝑁 ∧ ∃!𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
6358, 59, 62syl2an 493 . . . . . . . . . . . . . . 15 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
6454, 63mpbid 221 . . . . . . . . . . . . . 14 (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
6544, 64sylan 487 . . . . . . . . . . . . 13 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
6665eqcomd 2616 . . . . . . . . . . . 12 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → 𝑛 = (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
6766sneqd 4137 . . . . . . . . . . 11 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
68 eqeq1 2614 . . . . . . . . . . . 12 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
6968adantr 480 . . . . . . . . . . 11 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
7067, 69mpbird 246 . . . . . . . . . 10 ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑𝑥𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
7170ex 449 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
7271rexlimivw 3011 . . . . . . . 8 (∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
7337, 72syl 17 . . . . . . 7 ((𝑌 ∈ (𝑉 ∖ {𝑥}) ∧ ∀𝑏 ∈ (𝑉 ∖ {𝑥})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}) → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
7473expcom 450 . . . . . 6 (∀𝑏 ∈ (𝑉 ∖ {𝑥})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} → (𝑌 ∈ (𝑉 ∖ {𝑥}) → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})))
7532, 74syl 17 . . . . 5 ((𝑥𝑉 ∧ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}) → (𝑌 ∈ (𝑉 ∖ {𝑥}) → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})))
7675impancom 455 . . . 4 ((𝑥𝑉𝑌 ∈ (𝑉 ∖ {𝑥})) → (∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛} → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})))
7776imp 444 . . 3 (((𝑥𝑉𝑌 ∈ (𝑉 ∖ {𝑥})) ∧ ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑛𝑉 ((𝐺 NeighbVtx 𝑎) ∩ (𝐺 NeighbVtx 𝑏)) = {𝑛}) → ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
7824, 77mpcom 37 . 2 ((𝜑𝑥𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)})
792, 78syl5req 2657 1 ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∖ cdif 3537   ∩ cin 3539  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlem6  41474
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