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Mirrors > Home > MPE Home > Th. List > usgrexmpledg | Structured version Visualization version GIF version |
Description: The edges {0, 1}, {1, 2}, {2, 0}, {0, 3} of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 12-Jan-2020.) (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 |
---|---|
usgrexmpledg | ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgval 26552 | . 2 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | usgrexmpl.v | . . . . 5 ⊢ 𝑉 = (0...4) | |
3 | usgrexmpl.e | . . . . 5 ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 | |
4 | usgrexmpl.g | . . . . 5 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
5 | 2, 3, 4 | usgrexmpllem 26760 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
6 | 5 | simpri 478 | . . 3 ⊢ (iEdg‘𝐺) = 𝐸 |
7 | 6 | rneqi 5655 | . 2 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
8 | prex 5193 | . . . . . . 7 ⊢ {0, 1} ∈ V | |
9 | prex 5193 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
10 | 8, 9 | pm3.2i 463 | . . . . . 6 ⊢ ({0, 1} ∈ V ∧ {1, 2} ∈ V) |
11 | prex 5193 | . . . . . . 7 ⊢ {2, 0} ∈ V | |
12 | prex 5193 | . . . . . . 7 ⊢ {0, 3} ∈ V | |
13 | 11, 12 | pm3.2i 463 | . . . . . 6 ⊢ ({2, 0} ∈ V ∧ {0, 3} ∈ V) |
14 | 10, 13 | pm3.2i 463 | . . . . 5 ⊢ (({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) |
15 | usgrexmpldifpr 26758 | . . . . 5 ⊢ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3})) | |
16 | 14, 15 | pm3.2i 463 | . . . 4 ⊢ ((({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) ∧ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3}))) |
17 | 16, 3 | pm3.2i 463 | . . 3 ⊢ (((({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) ∧ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3}))) ∧ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉) |
18 | s4f1o 14148 | . . . 4 ⊢ ((({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) → ((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3})) → (𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 → 𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})))) | |
19 | 18 | imp31 410 | . . 3 ⊢ ((((({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) ∧ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3}))) ∧ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉) → 𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})) |
20 | dff1o5 6458 | . . . 4 ⊢ (𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝐸:dom 𝐸–1-1→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ∧ ran 𝐸 = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}))) | |
21 | 20 | simprbi 489 | . . 3 ⊢ (𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})) |
22 | 17, 19, 21 | mp2b 10 | . 2 ⊢ ran 𝐸 = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
23 | 1, 7, 22 | 3eqtri 2808 | 1 ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2969 Vcvv 3417 ∪ cun 3829 {cpr 4446 〈cop 4450 dom cdm 5411 ran crn 5412 –1-1→wf1 6190 –1-1-onto→wf1o 6192 ‘cfv 6193 (class class class)co 6982 0cc0 10341 1c1 10342 2c2 11501 3c3 11502 4c4 11503 ...cfz 12714 〈“cs4 14073 Vtxcvtx 26499 iEdgciedg 26500 Edgcedg 26550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-oadd 7915 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-3 11510 df-n0 11714 df-z 11800 df-uz 12065 df-fz 12715 df-fzo 12856 df-hash 13512 df-word 13679 df-concat 13740 df-s1 13765 df-s2 14078 df-s3 14079 df-s4 14080 df-vtx 26501 df-iedg 26502 df-edg 26551 |
This theorem is referenced by: (None) |
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