Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 6113 | . . . . 5 ⊢ (Vtx‘𝐻) ∈ V | |
2 | fvex 6113 | . . . . 5 ⊢ (Edg‘𝐻) ∈ V | |
3 | ausgr.1 | . . . . . 6 ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}} | |
4 | 3 | isausgr 40394 | . . . . 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 704 | . . . 4 ⊢ ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}) |
6 | edgaval 25794 | . . . . . 6 ⊢ (𝐻 ∈ 𝑊 → (Edg‘𝐻) = ran (iEdg‘𝐻)) | |
7 | 6 | sseq1d 3595 | . . . . 5 ⊢ (𝐻 ∈ 𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ↔ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})) |
8 | funfn 5833 | . . . . . . . . 9 ⊢ (Fun (iEdg‘𝐻) ↔ (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) | |
9 | 8 | biimpi 205 | . . . . . . . 8 ⊢ (Fun (iEdg‘𝐻) → (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) |
10 | 9 | 3ad2ant3 1077 | . . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ∧ Fun (iEdg‘𝐻)) → (iEdg‘𝐻) Fn dom (iEdg‘𝐻)) |
11 | simp2 1055 | . . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ∧ Fun (iEdg‘𝐻)) → ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}) | |
12 | df-f 5808 | . . . . . . 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})) | |
13 | 10, 11, 12 | sylanbrc 695 | . . . . . 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}) |
14 | 13 | 3exp 1256 | . . . . 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}))) |
15 | 7, 14 | sylbid 229 | . . . 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}))) |
16 | 5, 15 | syl5bi 231 | . . 3 ⊢ (𝐻 ∈ 𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → (Fun (iEdg‘𝐻) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))) |
17 | 16 | 3imp 1249 | . 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}) |
18 | eqid 2610 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻) | |
19 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻) | |
20 | 18, 19 | isumgrs 25762 | . . 3 ⊢ (𝐻 ∈ 𝑊 → (𝐻 ∈ UMGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})) |
21 | 20 | 3ad2ant1 1075 | . 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → (𝐻 ∈ UMGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})) |
22 | 17, 21 | mpbird 246 | 1 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ Fun (iEdg‘𝐻)) → 𝐻 ∈ UMGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 {copab 4642 dom cdm 5038 ran crn 5039 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 UMGraph cumgr 25748 Edgcedga 25792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-umgr 25750 df-edga 25793 |
This theorem is referenced by: (None) |
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