![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > usgrausgrb | Structured version Visualization version GIF version |
Description: The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
ausgr.1 | ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}} |
ausgrusgri.1 | ⊢ 𝑂 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓} |
Ref | Expression |
---|---|
usgrausgrb | ⊢ ((𝐻 ∈ 𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ausgr.1 | . . . . . 6 ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}} | |
2 | ausgrusgri.1 | . . . . . 6 ⊢ 𝑂 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓} | |
3 | 1, 2 | ausgrusgri 26284 | . . . . 5 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph) |
4 | 3 | 3exp 1113 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → ((iEdg‘𝐻) ∈ 𝑂 → 𝐻 ∈ USGraph))) |
5 | 4 | com23 86 | . . 3 ⊢ (𝐻 ∈ 𝑊 → ((iEdg‘𝐻) ∈ 𝑂 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → 𝐻 ∈ USGraph))) |
6 | 5 | imp 444 | . 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → 𝐻 ∈ USGraph)) |
7 | 1 | usgrausgri 26282 | . 2 ⊢ (𝐻 ∈ USGraph → (Vtx‘𝐻)𝐺(Edg‘𝐻)) |
8 | 6, 7 | impbid1 215 | 1 ⊢ ((𝐻 ∈ 𝑊 ∧ (iEdg‘𝐻) ∈ 𝑂) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ 𝐻 ∈ USGraph)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2140 {cab 2747 {crab 3055 ⊆ wss 3716 𝒫 cpw 4303 class class class wbr 4805 {copab 4865 dom cdm 5267 ran crn 5268 –1-1→wf1 6047 ‘cfv 6050 2c2 11283 ♯chash 13332 Vtxcvtx 26095 iEdgciedg 26096 Edgcedg 26160 USGraphcusgr 26265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-er 7914 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-n0 11506 df-z 11591 df-uz 11901 df-fz 12541 df-hash 13333 df-edg 26161 df-usgr 26267 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |