Theorem vtxdginducedm1lem3 29521

Description: Lemma 3 for vtxdginducedm1 29523: 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 29519	. . . 4 ⊢ (iEdg‘𝑆) = 𝑃
8	7, 5	eqtri 2758	. . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼)
9	8	fveq1i 6877	. 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻)
10		fvres 6895	. 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻))
11	9, 10	eqtrid 2782	1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∉ wnel 3036  {crab 3415   ∖ cdif 3923  {csn 4601  ⟨cop 4607  dom cdm 5654   ↾ cres 5656  ‘cfv 6531  Vtxcvtx 28975  iEdgciedg 28976

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fv 6539  df-2nd 7989  df-iedg 28978

This theorem is referenced by:  vtxdginducedm1  29523