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| Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for vtxdginducedm1 29561: 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 |
|---|---|
| vtxdginducedm1lem1 | ⊢ (iEdg‘𝑆) = 𝑃 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxdginducedm1.s | . . 3 ⊢ 𝑆 = 〈𝐾, 𝑃〉 | |
| 2 | 1 | fveq2i 6909 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘〈𝐾, 𝑃〉) |
| 3 | vtxdginducedm1.k | . . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁}) | |
| 4 | vtxdginducedm1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 5 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝑉 ∈ V |
| 6 | 5 | difexi 5330 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
| 7 | 3, 6 | eqeltri 2837 | . . 3 ⊢ 𝐾 ∈ V |
| 8 | vtxdginducedm1.p | . . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
| 9 | vtxdginducedm1.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 10 | 9 | fvexi 6920 | . . . . 5 ⊢ 𝐸 ∈ V |
| 11 | 10 | resex 6047 | . . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V |
| 12 | 8, 11 | eqeltri 2837 | . . 3 ⊢ 𝑃 ∈ V |
| 13 | 7, 12 | opiedgfvi 29027 | . 2 ⊢ (iEdg‘〈𝐾, 𝑃〉) = 𝑃 |
| 14 | 2, 13 | eqtri 2765 | 1 ⊢ (iEdg‘𝑆) = 𝑃 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∖ cdif 3948 {csn 4626 〈cop 4632 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 df-iedg 29016 |
| This theorem is referenced by: vtxdginducedm1lem2 29558 vtxdginducedm1lem3 29559 vtxdginducedm1fi 29562 finsumvtxdg2ssteplem4 29566 |
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