Theorem vtxdginducedm1lem2 27897

Description: Lemma 2 for vtxdginducedm1 27900: the domain of the edge function 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
vtxdginducedm1lem2	⊢ dom (iEdg‘𝑆) = 𝐼

Distinct variable group:   𝑖,𝐸

Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝐺(𝑖)   𝐼(𝑖)   𝐾(𝑖)   𝑁(𝑖)   𝑉(𝑖)

Proof of Theorem vtxdginducedm1lem2

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 27896	. . . 4 ⊢ (iEdg‘𝑆) = 𝑃
8	7, 5	eqtri 2768	. . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼)
9	8	dmeqi 5811	. 2 ⊢ dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐼)
10	4	ssrab3 4020	. . 3 ⊢ 𝐼 ⊆ dom 𝐸
11		ssdmres 5912	. . 3 ⊢ (𝐼 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐼) = 𝐼)
12	10, 11	mpbi 229	. 2 ⊢ dom (𝐸 ↾ 𝐼) = 𝐼
13	9, 12	eqtri 2768	1 ⊢ dom (iEdg‘𝑆) = 𝐼

Syntax hints:   = wceq 1542   ∉ wnel 3051  {crab 3070   ∖ cdif 3889   ⊆ wss 3892  {csn 4567  ⟨cop 4573  dom cdm 5589   ↾ cres 5591  ‘cfv 6431  Vtxcvtx 27356  iEdgciedg 27357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6389  df-fun 6433  df-fv 6439  df-2nd 7819  df-iedg 27359

This theorem is referenced by:  vtxdginducedm1  27900