Theorem vtxdginducedm1lem3 29743

Description: Lemma 3 for vtxdginducedm1 29745: 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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
vtxdginducedm1.p	⊢ 𝑃 = (𝐸 ↾ 𝐼)
vtxdginducedm1.s	⊢ 𝑆 = ⟨𝐾, 𝑃⟩

Assertion

Ref	Expression
vtxdginducedm1lem3	⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))

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

Step	Hyp	Ref	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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
5		vtxdginducedm1.p	. . . . 5 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
6		vtxdginducedm1.s	. . . . 5 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
7	1, 2, 3, 4, 5, 6	vtxdginducedm1lem1 29741	. . . 4 ⊢ (iEdg‘𝑆) = 𝑃
8	7, 5	eqtri 2786	. . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼)
9	8	fveq1i 6869	. 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻)
10		fvres 6887	. 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻))
11	9, 10	eqtrid 2810	1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143   ∉ wnel 3062  {crab 3415   ∖ cdif 3902  {csn 4583  ⟨cop 4589  dom cdm 5648   ↾ cres 5650  ‘cfv 6522  Vtxcvtx 29198  iEdgciedg 29199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-iota 6478  df-fun 6524  df-fv 6530  df-2nd 7972  df-iedg 29201

This theorem is referenced by:  vtxdginducedm1  29745