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Mirrors > Home > MPE Home > Th. List > opiedgfvi | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.) |
Ref | Expression |
---|---|
opvtxfvi.v | ⊢ 𝑉 ∈ V |
opvtxfvi.e | ⊢ 𝐸 ∈ V |
Ref | Expression |
---|---|
opiedgfvi | ⊢ (iEdg‘〈𝑉, 𝐸〉) = 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opvtxfvi.v | . 2 ⊢ 𝑉 ∈ V | |
2 | opvtxfvi.e | . 2 ⊢ 𝐸 ∈ V | |
3 | opiedgfv 26786 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (iEdg‘〈𝑉, 𝐸〉) = 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ‘cfv 6349 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-iota 6308 df-fun 6351 df-fv 6357 df-2nd 7684 df-iedg 26778 |
This theorem is referenced by: graop 26808 iedgvalsnop 26821 griedg0ssusgr 27041 uhgrspanop 27072 vtxdgop 27246 vtxdginducedm1lem1 27315 finsumvtxdg2size 27326 rgrusgrprc 27365 eupth2lem3 28009 konigsberglem1 28025 konigsberglem2 28026 konigsberglem3 28027 |
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