Theorem uhgrspan1lem1 28536

Description: Lemma 1 for uhgrspan1 28539. (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 6901	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5326	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		uhgrspan1.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	fvexi 6901	. . 3 ⊢ 𝐼 ∈ V
6	5	resex 6026	. 2 ⊢ (𝐼 ↾ 𝐹) ∈ V
7	3, 6	pm3.2i 472	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∉ wnel 3047  {crab 3433  Vcvv 3475   ∖ cdif 3943  {csn 4626  dom cdm 5674   ↾ cres 5676  ‘cfv 6539  Vtxcvtx 28235  iEdgciedg 28236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-sn 4627  df-pr 4629  df-uni 4907  df-res 5686  df-iota 6491  df-fv 6547

This theorem is referenced by:  uhgrspan1lem2  28537  uhgrspan1lem3  28538