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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 28249 | . 2 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
4 | usgrexmpl.g | . . . 4 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩ | |
5 | opex 5422 | . . . 4 ⊢ ⟨𝑉, 𝐸⟩ ∈ V | |
6 | 4, 5 | eqeltri 2830 | . . 3 ⊢ 𝐺 ∈ V |
7 | eqid 2733 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2733 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | isusgrs 28149 | . . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2})) |
10 | 1, 2, 4 | usgrexmpllem 28250 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) |
11 | simpr 486 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → (iEdg‘𝐺) = 𝐸) | |
12 | 11 | dmeqd 5862 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → dom (iEdg‘𝐺) = dom 𝐸) |
13 | pweq 4575 | . . . . . . . 8 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) | |
14 | 13 | adantr 482 | . . . . . . 7 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉) |
15 | 14 | rabeqdv 3421 | . . . . . 6 ⊢ (((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) → {𝑒 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
16 | 11, 12, 15 | f1eq123d 6777 | . . . . 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 | bitrdi 287 | . . 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 230 | 1 ⊢ 𝐺 ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 Vcvv 3444 𝒫 cpw 4561 {cpr 4589 ⟨cop 4593 dom cdm 5634 –1-1→wf1 6494 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 2c2 12213 3c3 12214 4c4 12215 ...cfz 13430 ♯chash 14236 ⟨“cs4 14738 Vtxcvtx 27989 iEdgciedg 27990 USGraphcusgr 28142 |
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-oadd 8417 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 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-4 12223 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-usgr 28144 |
This theorem is referenced by: (None) |
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