Theorem upgrres1lem1 27093

Description: Lemma 1 for upgrres1 27097. (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 6686	. . 3 ⊢ 𝑉 ∈ V
3	2	difexi 5234	. 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V
4		upgrres1.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
6	5	fvexi 6686	. . . 4 ⊢ 𝐸 ∈ V
7	4, 6	rabex2 5239	. . 3 ⊢ 𝐹 ∈ V
8		resiexg 7621	. . 3 ⊢ (𝐹 ∈ V → ( I ↾ 𝐹) ∈ V)
9	7, 8	ax-mp 5	. 2 ⊢ ( I ↾ 𝐹) ∈ V
10	3, 9	pm3.2i 473	1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V)

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∉ wnel 3125  {crab 3144  Vcvv 3496   ∖ cdif 3935  {csn 4569   I cid 5461   ↾ cres 5559  ‘cfv 6357  Vtxcvtx 26783  Edgcedg 26834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-res 5569  df-iota 6316  df-fv 6365

This theorem is referenced by:  upgrres1lem2  27095  upgrres1lem3  27096