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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for vtxdginducedm1 27327: 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 27323 | . . . 4 ⊢ (iEdg‘𝑆) = 𝑃 |
8 | 7, 5 | eqtri 2846 | . . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼) |
9 | 8 | fveq1i 6673 | . 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻) |
10 | fvres 6691 | . 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻)) | |
11 | 9, 10 | syl5eq 2870 | 1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 {crab 3144 ∖ cdif 3935 {csn 4569 〈cop 4575 dom cdm 5557 ↾ cres 5559 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-2nd 7692 df-iedg 26786 |
This theorem is referenced by: vtxdginducedm1 27327 |
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