Theorem upgrres1lem1 40528

Description: Lemma 1 for upgrres1 40532. (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		fvex 6113	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
3	1, 2	eqeltri 2684	. . 3 ⊢ 𝑉 ∈ V
4	3	difexi 4736	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
5		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
7		fvex 6113	. . . . 5 ⊢ (Edg‘𝐺) ∈ V
8	6, 7	eqeltri 2684	. . . 4 ⊢ 𝐸 ∈ V
9	5, 8	rabex2 4742	. . 3 ⊢ 𝐹 ∈ V
10		resiexg 6994	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
11	9, 10	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
12	4, 11	pm3.2i 470	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125   I cid 4948   ↾ cres 5040  ‘cfv 5804  Vtxcvtx 25673  Edgcedga 25792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-res 5050  df-iota 5768  df-fv 5812

This theorem is referenced by:  upgrres1lem2  40530  upgrres1lem3  40531