Theorem isubgrvtx 47867

Description: The vertices of an induced subgraph. (Contributed by AV, 12-May-2025.)

Hypothesis

Ref	Expression
isubgrvtx.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
isubgrvtx	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)

Proof of Theorem isubgrvtx

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgrvtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2729	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isisubgr 47862	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟩)
4	3	fveq2d 6862	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘⟨𝑆, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟩))
5	1	fvexi 6872	. . . 4 ⊢ 𝑉 ∈ V
6	5	ssex 5276	. . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ V)
7		fvexd 6873	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺) ∈ V)
8	7	resexd 5999	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ∈ V)
9		opvtxfv 28931	. . 3 ⊢ ((𝑆 ∈ V ∧ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ∈ V) → (Vtx‘⟨𝑆, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟩) = 𝑆)
10	6, 8, 9	syl2an2 686	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘⟨𝑆, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟩) = 𝑆)
11	4, 10	eqtrd 2764	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914  ⟨cop 4595  dom cdm 5638   ↾ cres 5640  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  iEdgciedg 28924   ISubGr cisubgr 47860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-vtx 28925  df-isubgr 47861

This theorem is referenced by:  isubgruhgr  47868  isubgrsubgr  47869  isubgrgrim  47929  isubgr3stgr  47974