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Mirrors > Home > MPE Home > Th. List > konigsbergvtx | Structured version Visualization version GIF version |
Description: The set of vertices of the KΓΆnigsberg graph πΊ. (Contributed by AV, 28-Feb-2021.) |
Ref | Expression |
---|---|
konigsberg.v | β’ π = (0...3) |
konigsberg.e | β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© |
konigsberg.g | β’ πΊ = β¨π, πΈβ© |
Ref | Expression |
---|---|
konigsbergvtx | β’ (VtxβπΊ) = (0...3) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg.g | . . . 4 β’ πΊ = β¨π, πΈβ© | |
2 | konigsberg.v | . . . . 5 β’ π = (0...3) | |
3 | konigsberg.e | . . . . 5 β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© | |
4 | 2, 3 | opeq12i 4875 | . . . 4 β’ β¨π, πΈβ© = β¨(0...3), β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ©β© |
5 | 1, 4 | eqtri 2753 | . . 3 β’ πΊ = β¨(0...3), β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ©β© |
6 | 5 | fveq2i 6893 | . 2 β’ (VtxβπΊ) = (Vtxββ¨(0...3), β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ©β©) |
7 | ovex 7446 | . . 3 β’ (0...3) β V | |
8 | s7cli 14863 | . . 3 β’ β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© β Word V | |
9 | opvtxfv 28856 | . . 3 β’ (((0...3) β V β§ β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© β Word V) β (Vtxββ¨(0...3), β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ©β©) = (0...3)) | |
10 | 7, 8, 9 | mp2an 690 | . 2 β’ (Vtxββ¨(0...3), β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ©β©) = (0...3) |
11 | 6, 10 | eqtri 2753 | 1 β’ (VtxβπΊ) = (0...3) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 Vcvv 3463 {cpr 4627 β¨cop 4631 βcfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 2c2 12292 3c3 12293 ...cfz 13511 Word cword 14491 β¨βcs7 14824 Vtxcvtx 28848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-s2 14826 df-s3 14827 df-s4 14828 df-s5 14829 df-s6 14830 df-s7 14831 df-vtx 28850 |
This theorem is referenced by: konigsbergumgr 30100 konigsberglem1 30101 konigsberglem2 30102 konigsberglem3 30103 |
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