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Theorem vtxdginducedm1lem2 26670
Description: Lemma 2 for vtxdginducedm1 26673: 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 𝐸𝑁 ∉ (𝐸𝑖)}
vtxdginducedm1.p 𝑃 = (𝐸𝐼)
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
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 26669 . . . 4 (iEdg‘𝑆) = 𝑃
87, 5eqtri 2792 . . 3 (iEdg‘𝑆) = (𝐸𝐼)
98dmeqi 5463 . 2 dom (iEdg‘𝑆) = dom (𝐸𝐼)
10 ssrab2 3834 . . . 4 {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)} ⊆ dom 𝐸
114, 10eqsstri 3782 . . 3 𝐼 ⊆ dom 𝐸
12 ssdmres 5561 . . 3 (𝐼 ⊆ dom 𝐸 ↔ dom (𝐸𝐼) = 𝐼)
1311, 12mpbi 220 . 2 dom (𝐸𝐼) = 𝐼
149, 13eqtri 2792 1 dom (iEdg‘𝑆) = 𝐼
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  wnel 3045  {crab 3064  cdif 3718  wss 3721  {csn 4314  cop 4320  dom cdm 5249  cres 5251  cfv 6031  Vtxcvtx 26094  iEdgciedg 26095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7315  df-iedg 26097
This theorem is referenced by:  vtxdginducedm1  26673
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