Theorem upgrres1lem1 29378

Description: Lemma 1 for upgrres1 29382. (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 6854	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5271	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6	5	fvexi 6854	. . . 4 ⊢ 𝐸 ∈ V
7	4, 6	rabex2 5282	. . 3 ⊢ 𝐹 ∈ V
8		resiexg 7863	. . 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 1542   ∈ wcel 2114   ∉ wnel 3036  {crab 3389  Vcvv 3429   ∖ cdif 3886  {csn 4567   I cid 5525   ↾ cres 5633  ‘cfv 6498  Vtxcvtx 29065  Edgcedg 29116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-res 5643  df-iota 6454  df-fv 6506

This theorem is referenced by:  upgrres1lem2  29380  upgrres1lem3  29381