Theorem clnbgrvtxedg 48800

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 48647	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ⊆ 𝑁)
5		sseq1 3965	. . 3 ⊢ (𝑥 = 𝐸 → (𝑥 ⊆ 𝑁 ↔ 𝐸 ⊆ 𝑁))
6		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
7	5, 6	elrab2 3657	. 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 2146  {crab 3419   ⊆ wss 3908  ‘cfv 6543  (class class class)co 7423  Edgcedg 29434  UHGraphcuhgr 29443   ClNeighbVtx cclnbgr 48624

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-edg 29435  df-uhgr 29445  df-clnbgr 48625

This theorem is used by:  grlimedgclnbgr  48801  grlimprclnbgredg  48803  grlimgrtrilem1  48807