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Theorem vtxdginducedm1lem3 29559
Description: Lemma 3 for vtxdginducedm1 29561: 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 29557 . . . 4 (iEdg‘𝑆) = 𝑃
87, 5eqtri 2765 . . 3 (iEdg‘𝑆) = (𝐸𝐼)
98fveq1i 6907 . 2 ((iEdg‘𝑆)‘𝐻) = ((𝐸𝐼)‘𝐻)
10 fvres 6925 . 2 (𝐻𝐼 → ((𝐸𝐼)‘𝐻) = (𝐸𝐻))
119, 10eqtrid 2789 1 (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wnel 3046  {crab 3436  cdif 3948  {csn 4626  cop 4632  dom cdm 5685  cres 5687  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-2nd 8015  df-iedg 29016
This theorem is referenced by:  vtxdginducedm1  29561
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