![]() |
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 28042 | . 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 28250 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
6 | 5 | simpri 487 | . . 3 ⊢ (iEdg‘𝐺) = 𝐸 |
7 | 6 | rneqi 5893 | . 2 ⊢ ran (iEdg‘𝐺) = ran 𝐸 |
8 | prex 5390 | . . . . . . 7 ⊢ {0, 1} ∈ V | |
9 | prex 5390 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
10 | 8, 9 | pm3.2i 472 | . . . . . 6 ⊢ ({0, 1} ∈ V ∧ {1, 2} ∈ V) |
11 | prex 5390 | . . . . . . 7 ⊢ {2, 0} ∈ V | |
12 | prex 5390 | . . . . . . 7 ⊢ {0, 3} ∈ V | |
13 | 11, 12 | pm3.2i 472 | . . . . . 6 ⊢ ({2, 0} ∈ V ∧ {0, 3} ∈ V) |
14 | 10, 13 | pm3.2i 472 | . . . . 5 ⊢ (({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) |
15 | usgrexmpldifpr 28248 | . . . . 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 472 | . . . 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 472 | . . 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 14813 | . . . 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 419 | . . 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 6794 | . . . 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 498 | . . 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 2765 | 1 ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 ∪ cun 3909 {cpr 4589 ⟨cop 4593 dom cdm 5634 ran crn 5635 –1-1→wf1 6494 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 2c2 12213 3c3 12214 4c4 12215 ...cfz 13430 ⟨“cs4 14738 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-concat 14465 df-s1 14490 df-s2 14743 df-s3 14744 df-s4 14745 df-vtx 27991 df-iedg 27992 df-edg 28041 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |