Theorem vtxdginducedm1lem2 29831

Description: Lemma 2 for vtxdginducedm1 29834: 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 29830	. . . 4 ⊢ (iEdg‘𝑆) = 𝑃
8	7, 5	eqtri 2792	. . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼)
9	8	dmeqi 5895	. 2 ⊢ dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐼)
10	4	ssrab3 4044	. . 3 ⊢ 𝐼 ⊆ dom 𝐸
11		ssdmres 6013	. . 3 ⊢ (𝐼 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐼) = 𝐼)
12	10, 11	mpbi 233	. 2 ⊢ dom (𝐸 ↾ 𝐼) = 𝐼
13	9, 12	eqtri 2792	1 ⊢ dom (iEdg‘𝑆) = 𝐼

Syntax hints:   = wceq 1567   ∉ wnel 3070  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  ⟨cop 4600  dom cdm 5662   ↾ cres 5664  ‘cfv 6537  Vtxcvtx 29287  iEdgciedg 29288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-2nd 7987  df-iedg 29290

This theorem is referenced by:  vtxdginducedm1  29834