Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrspan1lem1 Structured version   Visualization version   GIF version

Theorem uhgrspan1lem1 27202
 Description: Lemma 1 for uhgrspan1 27205. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
uhgrspan1.v 𝑉 = (Vtx‘𝐺)
uhgrspan1.i 𝐼 = (iEdg‘𝐺)
uhgrspan1.f 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
Assertion
Ref Expression
uhgrspan1lem1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)

Proof of Theorem uhgrspan1lem1
StepHypRef Expression
1 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
21fvexi 6677 . . 3 𝑉 ∈ V
32difexi 5202 . 2 (𝑉 ∖ {𝑁}) ∈ V
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
54fvexi 6677 . . 3 𝐼 ∈ V
65resex 5876 . 2 (𝐼𝐹) ∈ V
73, 6pm3.2i 474 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∉ wnel 3055  {crab 3074  Vcvv 3409   ∖ cdif 3857  {csn 4525  dom cdm 5528   ↾ cres 5530  ‘cfv 6340  Vtxcvtx 26901  iEdgciedg 26902 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-uni 4802  df-res 5540  df-iota 6299  df-fv 6348 This theorem is referenced by:  uhgrspan1lem2  27203  uhgrspan1lem3  27204
 Copyright terms: Public domain W3C validator