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Theorem vtxdginducedm1lem3 29574
Description: Lemma 3 for vtxdginducedm1 29576: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
Hypotheses
Ref Expression
vtxdginducedm1.v 𝑉 = (Vtx‘𝐺)
vtxdginducedm1.e 𝐸 = (iEdg‘𝐺)
vtxdginducedm1.k 𝐾 = (𝑉 ∖ {𝑁})
vtxdginducedm1.i 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
vtxdginducedm1.p 𝑃 = (𝐸𝐼)
vtxdginducedm1.s 𝑆 = ⟨𝐾, 𝑃
Assertion
Ref Expression
vtxdginducedm1lem3 (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
Distinct variable group:   𝑖,𝐸
Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝐺(𝑖)   𝐻(𝑖)   𝐼(𝑖)   𝐾(𝑖)   𝑁(𝑖)   𝑉(𝑖)

Proof of Theorem vtxdginducedm1lem3
StepHypRef Expression
1 vtxdginducedm1.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 vtxdginducedm1.e . . . . 5 𝐸 = (iEdg‘𝐺)
3 vtxdginducedm1.k . . . . 5 𝐾 = (𝑉 ∖ {𝑁})
4 vtxdginducedm1.i . . . . 5 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
5 vtxdginducedm1.p . . . . 5 𝑃 = (𝐸𝐼)
6 vtxdginducedm1.s . . . . 5 𝑆 = ⟨𝐾, 𝑃
71, 2, 3, 4, 5, 6vtxdginducedm1lem1 29572 . . . 4 (iEdg‘𝑆) = 𝑃
87, 5eqtri 2763 . . 3 (iEdg‘𝑆) = (𝐸𝐼)
98fveq1i 6908 . 2 ((iEdg‘𝑆)‘𝐻) = ((𝐸𝐼)‘𝐻)
10 fvres 6926 . 2 (𝐻𝐼 → ((𝐸𝐼)‘𝐻) = (𝐸𝐻))
119, 10eqtrid 2787 1 (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wnel 3044  {crab 3433  cdif 3960  {csn 4631  cop 4637  dom cdm 5689  cres 5691  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-2nd 8014  df-iedg 29031
This theorem is referenced by:  vtxdginducedm1  29576
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