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Mirrors > Home > MPE Home > Th. List > usgrexmpl | Structured version Visualization version GIF version |
Description: 𝐺 is a simple graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.) |
Ref | Expression |
---|---|
usgrexmpl.v | ⊢ 𝑉 = (0...4) |
usgrexmpl.e | ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩ |
usgrexmpl.g | ⊢ 𝐺 = ⟨𝑉, 𝐸⟩ |
Ref | Expression |
---|---|
usgrexmpl | ⊢ 𝐺 ∈ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrexmpl.v | . . 3 ⊢ 𝑉 = (0...4) | |
2 | usgrexmpl.e | . . 3 ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩ | |
3 | 1, 2 | usgrexmplef 29116 | . 2 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
4 | usgrexmpl.g | . . . 4 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩ | |
5 | opex 5460 | . . . 4 ⊢ ⟨𝑉, 𝐸⟩ ∈ V | |
6 | 4, 5 | eqeltri 2821 | . . 3 ⊢ 𝐺 ∈ V |
7 | eqid 2725 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2725 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | isusgrs 29013 | . . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2})) |
10 | 1, 2, 4 | usgrexmpllem 29117 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
11 | simpr 483 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → (iEdg‘𝐺) = 𝐸) | |
12 | 11 | dmeqd 5902 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → dom (iEdg‘𝐺) = dom 𝐸) |
13 | pweq 4612 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) | |
14 | 13 | adantr 479 | . . . . . . 7 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) |
15 | 14 | rabeqdv 3435 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → {𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
16 | 11, 12, 15 | f1eq123d 6826 | . . . . 5 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})) |
17 | 10, 16 | ax-mp 5 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
18 | 9, 17 | bitrdi 286 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})) |
19 | 6, 18 | ax-mp 5 | . 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
20 | 3, 19 | mpbir 230 | 1 ⊢ 𝐺 ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 𝒫 cpw 4598 {cpr 4626 ⟨cop 4630 dom cdm 5672 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 2c2 12297 3c3 12298 4c4 12299 ...cfz 13516 ♯chash 14321 ⟨“cs4 14826 Vtxcvtx 28853 iEdgciedg 28854 USGraphcusgr 29006 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-s4 14833 df-vtx 28855 df-iedg 28856 df-usgr 29008 |
This theorem is referenced by: (None) |
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