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Mirrors > Home > MPE Home > Th. List > ausgrumgri | Structured version Visualization version GIF version |
Description: If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
ausgr.1 | ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}} |
Ref | Expression |
---|---|
ausgrumgri | ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6898 | . . . . 5 ⊢ (Vtx‘𝐻) ∈ V | |
2 | fvex 6898 | . . . . 5 ⊢ (Edg‘𝐻) ∈ V | |
3 | ausgr.1 | . . . . . 6 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}} | |
4 | 3 | isausgr 28932 | . . . . 5 ⊢ (((Vtx‘𝐻) ∈ V ∧ (Edg‘𝐻) ∈ V) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
5 | 1, 2, 4 | mp2an 689 | . . . 4 ⊢ ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
6 | edgval 28817 | . . . . . . 7 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻) | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝐻 ∈ 𝑊 → (Edg‘𝐻) = ran (iEdg‘𝐻)) |
8 | 7 | sseq1d 4008 | . . . . 5 ⊢ (𝐻 ∈ 𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} ↔ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
9 | funfn 6572 | . . . . . . . . 9 ⊢ (Fun (iEdg‘𝐻) ↔ (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) | |
10 | 9 | biimpi 215 | . . . . . . . 8 ⊢ (Fun (iEdg‘𝐻) → (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) |
11 | 10 | 3ad2ant3 1132 | . . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} ∧ Fun (iEdg‘𝐻)) → (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) |
12 | simp2 1134 | . . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} ∧ Fun (iEdg‘𝐻)) → ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) | |
13 | df-f 6541 | . . . . . . 7 ⊢ ((iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} ↔ ((iEdg‘𝐻) Fn dom (iEdg‘𝐻) ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) | |
14 | 11, 12, 13 | sylanbrc 582 | . . . . . 6 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} ∧ Fun (iEdg‘𝐻)) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
15 | 14 | 3exp 1116 | . . . . 5 ⊢ (𝐻 ∈ 𝑊 → (ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} → (Fun (iEdg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}))) |
16 | 8, 15 | sylbid 239 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2} → (Fun (iEdg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}))) |
17 | 5, 16 | biimtrid 241 | . . 3 ⊢ (𝐻 ∈ 𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → (Fun (iEdg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}))) |
18 | 17 | 3imp 1108 | . 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2}) |
19 | eqid 2726 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻) | |
20 | eqid 2726 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻) | |
21 | 19, 20 | isumgrs 28864 | . . 3 ⊢ (𝐻 ∈ 𝑊 → (𝐻 ∈ UMGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
22 | 21 | 3ad2ant1 1130 | . 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → (𝐻 ∈ UMGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (♯‘𝑥) = 2})) |
23 | 18, 22 | mpbird 257 | 1 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 {copab 5203 dom cdm 5669 ran crn 5670 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 2c2 12271 ♯chash 14295 Vtxcvtx 28764 iEdgciedg 28765 Edgcedg 28815 UMGraphcumgr 28849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 df-edg 28816 df-umgr 28851 |
This theorem is referenced by: (None) |
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