Theorem clnbgrvtxedg 48024

Description: An edge 𝐸 containing a vertex 𝐴 is an edge in the closed neighborhood of this vertex 𝐴. (Contributed by AV, 25-Dec-2025.)

Hypotheses

Ref	Expression
clnbgrvtxedg.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
clnbgrvtxedg.i	⊢ 𝐼 = (Edg‘𝐺)
clnbgrvtxedg.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}

Assertion

Ref	Expression
clnbgrvtxedg	⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼   𝑥,𝑁

Allowed substitution hints:   𝐴(𝑥)   𝐺(𝑥)   𝐾(𝑥)

Proof of Theorem clnbgrvtxedg

Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐼)
2		clnbgrvtxedg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐺)
3		clnbgrvtxedg.n	. . 3 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
4	2, 3	clnbgrssedg 47871	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ⊆ 𝑁)
5		sseq1 3960	. . 3 ⊢ (𝑥 = 𝐸 → (𝑥 ⊆ 𝑁 ↔ 𝐸 ⊆ 𝑁))
6		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
7	5, 6	elrab2 3650	. 2 ⊢ (𝐸 ∈ 𝐾 ↔ (𝐸 ∈ 𝐼 ∧ 𝐸 ⊆ 𝑁))
8	1, 4, 7	sylanbrc 583	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3902  ‘cfv 6481  (class class class)co 7346  Edgcedg 29023  UHGraphcuhgr 29032   ClNeighbVtx cclnbgr 47848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-edg 29024  df-uhgr 29034  df-clnbgr 47849

This theorem is referenced by:  grlimedgclnbgr  48025  grlimprclnbgredg  48027  grlimgrtrilem1  48031