Theorem clnbgrvtxedg 49365

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 1155	. 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 49212	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ⊆ 𝑁)
5		sseq1 3938	. . 3 ⊢ (𝑥 = 𝐸 → (𝑥 ⊆ 𝑁 ↔ 𝐸 ⊆ 𝑁))
6		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
7	5, 6	elrab2 3652	. 2 ⊢ (𝐸 ∈ 𝐾 ↔ (𝐸 ∈ 𝐼 ∧ 𝐸 ⊆ 𝑁))
8	1, 4, 7	sylanbrc 595	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900  ‘cfv 6566  (class class class)co 7451  Edgcedg 29682  UHGraphcuhgr 29691   ClNeighbVtx cclnbgr 49189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-edg 29683  df-uhgr 29693  df-clnbgr 49190

This theorem is used by:  grlimedgclnbgr  49366  grlimprclnbgredg  49368  grlimgrtrilem1  49372