Theorem upgrres1lem1 26089

Description: Lemma 1 for upgrres1 26093. (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 6158	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
3	1, 2	eqeltri 2694	. . 3 ⊢ 𝑉 ∈ V
4	3	difexi 4769	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
5		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
6		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
7		fvex 6158	. . . . 5 ⊢ (Edg‘𝐺) ∈ V
8	6, 7	eqeltri 2694	. . . 4 ⊢ 𝐸 ∈ V
9	5, 8	rabex2 4775	. . 3 ⊢ 𝐹 ∈ V
10		resiexg 7049	. . 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 1480   ∈ wcel 1987   ∉ wnel 2893  {crab 2911  Vcvv 3186   ∖ cdif 3552  {csn 4148   I cid 4984   ↾ cres 5076  ‘cfv 5847  Vtxcvtx 25774  Edgcedg 25839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-res 5086  df-iota 5810  df-fv 5855

This theorem is referenced by:  upgrres1lem2  26091  upgrres1lem3  26092