Theorem clnbgrvtxedg 48492

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 1143	. 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 48339	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ⊆ 𝑁)
5		sseq1 3947	. . 3 ⊢ (𝑥 = 𝐸 → (𝑥 ⊆ 𝑁 ↔ 𝐸 ⊆ 𝑁))
6		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
7	5, 6	elrab2 3639	. 2 ⊢ (𝐸 ∈ 𝐾 ↔ (𝐸 ∈ 𝐼 ∧ 𝐸 ⊆ 𝑁))
8	1, 4, 7	sylanbrc 589	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ 𝐾)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3392   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Edgcedg 29141  UHGraphcuhgr 29150   ClNeighbVtx cclnbgr 48316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-edg 29142  df-uhgr 29152  df-clnbgr 48317

This theorem is referenced by:  grlimedgclnbgr  48493  grlimprclnbgredg  48495  grlimgrtrilem1  48499