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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for vtxdginducedm1 27319: 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.) |
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 |
---|---|
vtxdginducedm1lem2 | ⊢ dom (iEdg‘𝑆) = 𝐼 |
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 27315 | . . . 4 ⊢ (iEdg‘𝑆) = 𝑃 |
8 | 7, 5 | eqtri 2844 | . . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼) |
9 | 8 | dmeqi 5767 | . 2 ⊢ dom (iEdg‘𝑆) = dom (𝐸 ↾ 𝐼) |
10 | 4 | ssrab3 4056 | . . 3 ⊢ 𝐼 ⊆ dom 𝐸 |
11 | ssdmres 5870 | . . 3 ⊢ (𝐼 ⊆ dom 𝐸 ↔ dom (𝐸 ↾ 𝐼) = 𝐼) | |
12 | 10, 11 | mpbi 232 | . 2 ⊢ dom (𝐸 ↾ 𝐼) = 𝐼 |
13 | 9, 12 | eqtri 2844 | 1 ⊢ dom (iEdg‘𝑆) = 𝐼 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∉ wnel 3123 {crab 3142 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 〈cop 4566 dom cdm 5549 ↾ cres 5551 ‘cfv 6349 Vtxcvtx 26775 iEdgciedg 26776 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fv 6357 df-2nd 7684 df-iedg 26778 |
This theorem is referenced by: vtxdginducedm1 27319 |
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