Theorem upgrres1lem1 29293

Description: Lemma 1 for upgrres1 29297. (Contributed by AV, 7-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	⊢ 𝑉 = (Vtx‘𝐺)
upgrres1.e	⊢ 𝐸 = (Edg‘𝐺)
upgrres1.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}

Assertion

Ref	Expression
upgrres1lem1	⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Allowed substitution hint:   𝐹(𝑒)

Proof of Theorem upgrres1lem1

Step	Hyp	Ref	Expression
1		upgrres1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6895	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5305	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6	5	fvexi 6895	. . . 4 ⊢ 𝐸 ∈ V
7	4, 6	rabex2 5316	. . 3 ⊢ 𝐹 ∈ V
8		resiexg 7913	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
9	7, 8	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
10	3, 9	pm3.2i 470	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∉ wnel 3037  {crab 3420  Vcvv 3464   ∖ cdif 3928  {csn 4606   I cid 5552   ↾ cres 5661  ‘cfv 6536  Vtxcvtx 28980  Edgcedg 29031

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-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-res 5671  df-iota 6489  df-fv 6544

This theorem is referenced by:  upgrres1lem2  29295  upgrres1lem3  29296