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Mirrors > Home > MPE Home > Th. List > usgrexmpl | Structured version Visualization version GIF version |
Description: 𝐺 is a simple graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (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 |
---|---|
usgrexmpl | ⊢ 𝐺 ∈ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrexmpl.v | . . 3 ⊢ 𝑉 = (0...4) | |
2 | usgrexmpl.e | . . 3 ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 | |
3 | 1, 2 | usgrexmplef 27049 | . 2 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
4 | usgrexmpl.g | . . . 4 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
5 | opex 5321 | . . . 4 ⊢ 〈𝑉, 𝐸〉 ∈ V | |
6 | 4, 5 | eqeltri 2886 | . . 3 ⊢ 𝐺 ∈ V |
7 | eqid 2798 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2798 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | isusgrs 26949 | . . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2})) |
10 | 1, 2, 4 | usgrexmpllem 27050 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
11 | simpr 488 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → (iEdg‘𝐺) = 𝐸) | |
12 | 11 | dmeqd 5738 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → dom (iEdg‘𝐺) = dom 𝐸) |
13 | pweq 4513 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) | |
14 | 13 | adantr 484 | . . . . . . 7 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) |
15 | 14 | rabeqdv 3432 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → {𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
16 | 11, 12, 15 | f1eq123d 6583 | . . . . 5 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})) |
17 | 10, 16 | ax-mp 5 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
18 | 9, 17 | syl6bb 290 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})) |
19 | 6, 18 | ax-mp 5 | . 2 ⊢ (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
20 | 3, 19 | mpbir 234 | 1 ⊢ 𝐺 ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 𝒫 cpw 4497 {cpr 4527 〈cop 4531 dom cdm 5519 –1-1→wf1 6321 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 2c2 11680 3c3 11681 4c4 11682 ...cfz 12885 ♯chash 13686 〈“cs4 14196 Vtxcvtx 26789 iEdgciedg 26790 USGraphcusgr 26942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-s4 14203 df-vtx 26791 df-iedg 26792 df-usgr 26944 |
This theorem is referenced by: (None) |
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