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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for vtxdginducedm1 26439: 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 6194 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘〈𝐾, 𝑃〉) |
3 | vtxdginducedm1.k | . . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁}) | |
4 | vtxdginducedm1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | fvexi 6202 | . . . . 5 ⊢ 𝑉 ∈ V |
6 | 5 | difexi 4809 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
7 | 3, 6 | eqeltri 2697 | . . 3 ⊢ 𝐾 ∈ V |
8 | vtxdginducedm1.p | . . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
9 | vtxdginducedm1.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
10 | 9 | fvexi 6202 | . . . . 5 ⊢ 𝐸 ∈ V |
11 | 10 | resex 5443 | . . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V |
12 | 8, 11 | eqeltri 2697 | . . 3 ⊢ 𝑃 ∈ V |
13 | 7, 12 | opiedgfvi 25890 | . 2 ⊢ (iEdg‘〈𝐾, 𝑃〉) = 𝑃 |
14 | 2, 13 | eqtri 2644 | 1 ⊢ (iEdg‘𝑆) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∉ wnel 2897 {crab 2916 Vcvv 3200 ∖ cdif 3571 {csn 4177 〈cop 4183 dom cdm 5114 ↾ cres 5116 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-2nd 7169 df-iedg 25877 |
This theorem is referenced by: vtxdginducedm1lem2 26436 vtxdginducedm1lem3 26437 vtxdginducedm1fi 26440 finsumvtxdg2ssteplem4 26444 |
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