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Mirrors > Home > MPE Home > Th. List > konigsbergumgr | Structured version Visualization version GIF version |
Description: The Königsberg graph 𝐺 is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.) |
Ref | Expression |
---|---|
konigsberg.v | ⊢ 𝑉 = (0...3) |
konigsberg.e | ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
konigsberg.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
Ref | Expression |
---|---|
konigsbergumgr | ⊢ 𝐺 ∈ UMGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg.v | . . 3 ⊢ 𝑉 = (0...3) | |
2 | konigsberg.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
3 | konigsberg.g | . . 3 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
4 | 1, 2, 3 | konigsbergiedgw 28027 | . 2 ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
5 | opex 5356 | . . . 4 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
6 | 3, 5 | eqeltri 2909 | . . 3 ⊢ 𝐺 ∈ V |
7 | s7cli 14247 | . . . 4 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word V | |
8 | 2, 7 | eqeltri 2909 | . . 3 ⊢ 𝐸 ∈ Word V |
9 | 1, 2, 3 | konigsbergvtx 28025 | . . . . 5 ⊢ (Vtx‘𝐺) = (0...3) |
10 | 1, 9 | eqtr4i 2847 | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) |
11 | 1, 2, 3 | konigsbergiedg 28026 | . . . . 5 ⊢ (iEdg‘𝐺) = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
12 | 2, 11 | eqtr4i 2847 | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) |
13 | 10, 12 | wrdumgr 26882 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐸 ∈ Word V) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
14 | 6, 8, 13 | mp2an 690 | . 2 ⊢ (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
15 | 4, 14 | mpbir 233 | 1 ⊢ 𝐺 ∈ UMGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 𝒫 cpw 4539 {cpr 4569 〈cop 4573 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 2c2 11693 3c3 11694 ...cfz 12893 ♯chash 13691 Word cword 13862 〈“cs7 14208 Vtxcvtx 26781 iEdgciedg 26782 UMGraphcumgr 26866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-s4 14212 df-s5 14213 df-s6 14214 df-s7 14215 df-vtx 26783 df-iedg 26784 df-umgr 26868 |
This theorem is referenced by: konigsberg 28036 |
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