![]() |
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 28901 | . 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 29112 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
6 | 5 | simpri 484 | . . 3 ⊢ (iEdg‘𝐺) = 𝐸 |
7 | 6 | rneqi 5934 | . 2 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
8 | prex 5429 | . . . . . . 7 ⊢ {0, 1} ∈ V | |
9 | prex 5429 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
10 | 8, 9 | pm3.2i 469 | . . . . . 6 ⊢ ({0, 1} ∈ V ∧ {1, 2} ∈ V) |
11 | prex 5429 | . . . . . . 7 ⊢ {2, 0} ∈ V | |
12 | prex 5429 | . . . . . . 7 ⊢ {0, 3} ∈ V | |
13 | 11, 12 | pm3.2i 469 | . . . . . 6 ⊢ ({2, 0} ∈ V ∧ {0, 3} ∈ V) |
14 | 10, 13 | pm3.2i 469 | . . . . 5 ⊢ (({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) |
15 | usgrexmpldifpr 29110 | . . . . 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 469 | . . . 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 469 | . . 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 14896 | . . . 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 416 | . . 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 6841 | . . . 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 495 | . . 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 2757 | 1 ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 ∪ cun 3939 {cpr 4627 ⟨cop 4631 dom cdm 5673 ran crn 5674 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 2c2 12292 3c3 12293 4c4 12294 ...cfz 13511 ⟨“cs4 14821 Vtxcvtx 28848 iEdgciedg 28849 Edgcedg 28899 |
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-2 12300 df-3 12301 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-vtx 28850 df-iedg 28851 df-edg 28900 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |