Theorem upgrres1lem1 26371

Description: Lemma 1 for upgrres1 26375. (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 6350	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
3	1, 2	eqeltri 2823	. . 3 ⊢ 𝑉 ∈ V
4	3	difexi 4949	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
5		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
7		fvex 6350	. . . . 5 ⊢ (Edg‘𝐺) ∈ V
8	6, 7	eqeltri 2823	. . . 4 ⊢ 𝐸 ∈ V
9	5, 8	rabex2 4954	. . 3 ⊢ 𝐹 ∈ V
10		resiexg 7255	. . 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 1620   ∈ wcel 2127   ∉ wnel 3023  {crab 3042  Vcvv 3328   ∖ cdif 3700  {csn 4309   I cid 5161   ↾ cres 5256  ‘cfv 6037  Vtxcvtx 26044  Edgcedg 26109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-res 5266  df-iota 6000  df-fv 6045

This theorem is referenced by:  upgrres1lem2  26373  upgrres1lem3  26374