Theorem upgrres1lem1 26201

Description: Lemma 1 for upgrres1 26205. (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 6201	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
3	1, 2	eqeltri 2697	. . 3 ⊢ 𝑉 ∈ V
4	3	difexi 4809	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
5		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
7		fvex 6201	. . . . 5 ⊢ (Edg‘𝐺) ∈ V
8	6, 7	eqeltri 2697	. . . 4 ⊢ 𝐸 ∈ V
9	5, 8	rabex2 4815	. . 3 ⊢ 𝐹 ∈ V
10		resiexg 7102	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
11	9, 10	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
12	4, 11	pm3.2i 471	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  {crab 2916  Vcvv 3200   ∖ cdif 3571  {csn 4177   I cid 5023   ↾ cres 5116  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  upgrres1lem2  26203  upgrres1lem3  26204