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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 28836 | . 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 29047 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
6 | 5 | simpri 485 | . . 3 ⊢ (iEdg‘𝐺) = 𝐸 |
7 | 6 | rneqi 5933 | . 2 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
8 | prex 5428 | . . . . . . 7 ⊢ {0, 1} ∈ V | |
9 | prex 5428 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
10 | 8, 9 | pm3.2i 470 | . . . . . 6 ⊢ ({0, 1} ∈ V ∧ {1, 2} ∈ V) |
11 | prex 5428 | . . . . . . 7 ⊢ {2, 0} ∈ V | |
12 | prex 5428 | . . . . . . 7 ⊢ {0, 3} ∈ V | |
13 | 11, 12 | pm3.2i 470 | . . . . . 6 ⊢ ({2, 0} ∈ V ∧ {0, 3} ∈ V) |
14 | 10, 13 | pm3.2i 470 | . . . . 5 ⊢ (({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) |
15 | usgrexmpldifpr 29045 | . . . . 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 470 | . . . 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 470 | . . 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 14887 | . . . 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 417 | . . 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 6842 | . . . 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 496 | . . 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 2759 | 1 ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 ∪ cun 3942 {cpr 4626 ⟨cop 4630 dom cdm 5672 ran crn 5673 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 0cc0 11124 1c1 11125 2c2 12283 3c3 12284 4c4 12285 ...cfz 13502 ⟨“cs4 14812 Vtxcvtx 28783 iEdgciedg 28784 Edgcedg 28834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-hash 14308 df-word 14483 df-concat 14539 df-s1 14564 df-s2 14817 df-s3 14818 df-s4 14819 df-vtx 28785 df-iedg 28786 df-edg 28835 |
This theorem is referenced by: (None) |
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