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

Theorem uhgrspan1lem1 26119
Description: Lemma 1 for uhgrspan1 26122. (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)
2 fvex 6168 . . . 4 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2694 . . 3 𝑉 ∈ V
43difexi 4779 . 2 (𝑉 ∖ {𝑁}) ∈ V
5 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
6 fvex 6168 . . . 4 (iEdg‘𝐺) ∈ V
75, 6eqeltri 2694 . . 3 𝐼 ∈ V
87resex 5412 . 2 (𝐼𝐹) ∈ V
94, 8pm3.2i 471 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wnel 2893  {crab 2912  Vcvv 3190  cdif 3557  {csn 4155  dom cdm 5084  cres 5086  cfv 5857  Vtxcvtx 25808  iEdgciedg 25809
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-pr 4158  df-uni 4410  df-res 5096  df-iota 5820  df-fv 5865
This theorem is referenced by:  uhgrspan1lem2  26120  uhgrspan1lem3  26121
  Copyright terms: Public domain W3C validator