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Theorem vtxdginducedm1lem3 28197
Description: Lemma 3 for vtxdginducedm1 28199: 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 𝐸𝑁 ∉ (𝐸𝑖)}
vtxdginducedm1.p 𝑃 = (𝐸𝐼)
vtxdginducedm1.s 𝑆 = ⟨𝐾, 𝑃
Assertion
Ref Expression
vtxdginducedm1lem3 (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
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
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 28195 . . . 4 (iEdg‘𝑆) = 𝑃
87, 5eqtri 2764 . . 3 (iEdg‘𝑆) = (𝐸𝐼)
98fveq1i 6826 . 2 ((iEdg‘𝑆)‘𝐻) = ((𝐸𝐼)‘𝐻)
10 fvres 6844 . 2 (𝐻𝐼 → ((𝐸𝐼)‘𝐻) = (𝐸𝐻))
119, 10eqtrid 2788 1 (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wnel 3046  {crab 3403  cdif 3895  {csn 4573  cop 4579  dom cdm 5620  cres 5622  cfv 6479  Vtxcvtx 27655  iEdgciedg 27656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-iota 6431  df-fun 6481  df-fv 6487  df-2nd 7900  df-iedg 27658
This theorem is referenced by:  vtxdginducedm1  28199
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