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Theorem uhgrspan1lem3 26104
Description: Lemma 3 for uhgrspan1 26105. (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)}
uhgrspan1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
Assertion
Ref Expression
uhgrspan1lem3 (iEdg‘𝑆) = (𝐼𝐹)

Proof of Theorem uhgrspan1lem3
StepHypRef Expression
1 uhgrspan1.s . . 3 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
21fveq2i 6156 . 2 (iEdg‘𝑆) = (iEdg‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩)
3 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
5 uhgrspan1.f . . . 4 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
63, 4, 5uhgrspan1lem1 26102 . . 3 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
7 opiedgfv 25804 . . 3 (((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V) → (iEdg‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝐼𝐹))
86, 7ax-mp 5 . 2 (iEdg‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝐼𝐹)
92, 8eqtri 2643 1 (iEdg‘𝑆) = (𝐼𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wnel 2893  {crab 2911  Vcvv 3189  cdif 3556  {csn 4153  cop 4159  dom cdm 5079  cres 5081  cfv 5852  Vtxcvtx 25791  iEdgciedg 25792
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fv 5860  df-2nd 7121  df-iedg 25794
This theorem is referenced by:  uhgrspan1  26105
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