Theorem uhgrspan1lem1 29458

Description: Lemma 1 for uhgrspan1 29461. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan1.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspan1.i	⊢ 𝐼 = (iEdg‘𝐺)
uhgrspan1.f	⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)}

Assertion

Ref	Expression
uhgrspan1lem1	⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V)

Proof of Theorem uhgrspan1lem1

Step	Hyp	Ref	Expression
1		uhgrspan1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6876	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5283	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		uhgrspan1.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	fvexi 6876	. . 3 ⊢ 𝐼 ∈ V
6	5	resex 6011	. 2 ⊢ (𝐼 ↾ 𝐹) ∈ V
7	3, 6	pm3.2i 474	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  {crab 3413  Vcvv 3453   ∖ cdif 3899  {csn 4579  dom cdm 5643   ↾ cres 5645  ‘cfv 6516  Vtxcvtx 29154  iEdgciedg 29155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-sn 4580  df-pr 4582  df-uni 4863  df-res 5655  df-iota 6472  df-fv 6524

This theorem is referenced by:  uhgrspan1lem2  29459  uhgrspan1lem3  29460