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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for vtxdginducedm1 29576: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.) |
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〉 |
Ref | Expression |
---|---|
vtxdginducedm1lem3 | ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) |
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 29572 | . . . 4 ⊢ (iEdg‘𝑆) = 𝑃 |
8 | 7, 5 | eqtri 2763 | . . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼) |
9 | 8 | fveq1i 6908 | . 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻) |
10 | fvres 6926 | . 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻)) | |
11 | 9, 10 | eqtrid 2787 | 1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 ∖ cdif 3960 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 Vtxcvtx 29028 iEdgciedg 29029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-2nd 8014 df-iedg 29031 |
This theorem is referenced by: vtxdginducedm1 29576 |
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