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Mirrors > Home > MPE Home > Th. List > opvtxfvi | Structured version Visualization version GIF version |
Description: The set of vertices 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 |
---|---|
opvtxfvi | ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opvtxfvi.v | . 2 ⊢ 𝑉 ∈ V | |
2 | opvtxfvi.e | . 2 ⊢ 𝐸 ∈ V | |
3 | opvtxfv 26716 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 ‘cfv 6348 Vtxcvtx 26708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 df-vtx 26710 |
This theorem is referenced by: graop 26741 vtxvalsnop 26753 uhgrspanop 27005 fusgrfis 27039 cusgrsize 27163 fusgrmaxsize 27173 vtxdgop 27179 vtxdginducedm1 27252 vtxdginducedm1fi 27253 finsumvtxdg2ssteplem4 27257 finsumvtxdg2size 27259 eupth2lem3 27942 konigsberglem1 27958 konigsberglem2 27959 konigsberglem3 27960 |
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