Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 26761 | . 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 26969 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
6 | 5 | simpri 486 | . . 3 ⊢ (iEdg‘𝐺) = 𝐸 |
7 | 6 | rneqi 5800 | . 2 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
8 | prex 5323 | . . . . . . 7 ⊢ {0, 1} ∈ V | |
9 | prex 5323 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
10 | 8, 9 | pm3.2i 471 | . . . . . 6 ⊢ ({0, 1} ∈ V ∧ {1, 2} ∈ V) |
11 | prex 5323 | . . . . . . 7 ⊢ {2, 0} ∈ V | |
12 | prex 5323 | . . . . . . 7 ⊢ {0, 3} ∈ V | |
13 | 11, 12 | pm3.2i 471 | . . . . . 6 ⊢ ({2, 0} ∈ V ∧ {0, 3} ∈ V) |
14 | 10, 13 | pm3.2i 471 | . . . . 5 ⊢ (({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) |
15 | usgrexmpldifpr 26967 | . . . . 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 471 | . . . 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 471 | . . 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 14268 | . . . 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 418 | . . 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 6617 | . . . 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 497 | . . 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 2845 | 1 ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∪ cun 3931 {cpr 4559 〈cop 4563 dom cdm 5548 ran crn 5549 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 2c2 11680 3c3 11681 4c4 11682 ...cfz 12880 〈“cs4 14193 Vtxcvtx 26708 iEdgciedg 26709 Edgcedg 26759 |
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-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-s4 14200 df-vtx 26710 df-iedg 26711 df-edg 26760 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |