Theorem upgrres1lem1 27676

Description: Lemma 1 for upgrres1 27680. (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 6788	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5252	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6	5	fvexi 6788	. . . 4 ⊢ 𝐸 ∈ V
7	4, 6	rabex2 5258	. . 3 ⊢ 𝐹 ∈ V
8		resiexg 7761	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
9	7, 8	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
10	3, 9	pm3.2i 471	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∉ wnel 3049  {crab 3068  Vcvv 3432   ∖ cdif 3884  {csn 4561   I cid 5488   ↾ cres 5591  ‘cfv 6433  Vtxcvtx 27366  Edgcedg 27417

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-res 5601  df-iota 6391  df-fv 6441

This theorem is referenced by:  upgrres1lem2  27678  upgrres1lem3  27679