Theorem upgrres1lem1 29400

Description: Lemma 1 for upgrres1 29404. (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 6845	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5261	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6	5	fvexi 6845	. . . 4 ⊢ 𝐸 ∈ V
7	4, 6	rabex2 5272	. . 3 ⊢ 𝐹 ∈ V
8		resiexg 7856	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
9	7, 8	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
10	3, 9	pm3.2i 472	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ∉ wnel 3040  {crab 3393  Vcvv 3433   ∖ cdif 3882  {csn 4558   I cid 5515   ↾ cres 5623  ‘cfv 6489  Vtxcvtx 29087  Edgcedg 29138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-res 5633  df-iota 6445  df-fv 6497

This theorem is referenced by:  upgrres1lem2  29402  upgrres1lem3  29403