Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ausgrusgri Structured version   Visualization version   GIF version

Theorem ausgrusgri 40398
 Description: The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
Hypotheses
Ref Expression
ausgr.1 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}
ausgrusgri.1 𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}
Assertion
Ref Expression
ausgrusgri ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph )
Distinct variable groups:   𝑣,𝑒,𝑥,𝐻   𝑓,𝐻   𝑥,𝑊
Allowed substitution hints:   𝐺(𝑥,𝑣,𝑒,𝑓)   𝑂(𝑥,𝑣,𝑒,𝑓)   𝑊(𝑣,𝑒,𝑓)

Proof of Theorem ausgrusgri
StepHypRef 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}}
43isausgr 40394 . . . . 5 (((Vtx‘𝐻) ∈ V ∧ (Edg‘𝐻) ∈ V) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
51, 2, 4mp2an 704 . . . 4 ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
6 edgaval 25794 . . . . . 6 (𝐻𝑊 → (Edg‘𝐻) = ran (iEdg‘𝐻))
76sseq1d 3595 . . . . 5 (𝐻𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ↔ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
8 ausgrusgri.1 . . . . . . . . . 10 𝑂 = {𝑓𝑓:dom 𝑓1-1→ran 𝑓}
98eleq2i 2680 . . . . . . . . 9 ((iEdg‘𝐻) ∈ 𝑂 ↔ (iEdg‘𝐻) ∈ {𝑓𝑓:dom 𝑓1-1→ran 𝑓})
10 fvex 6113 . . . . . . . . . 10 (iEdg‘𝐻) ∈ V
11 id 22 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → 𝑓 = (iEdg‘𝐻))
12 dmeq 5246 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → dom 𝑓 = dom (iEdg‘𝐻))
13 rneq 5272 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → ran 𝑓 = ran (iEdg‘𝐻))
1411, 12, 13f1eq123d 6044 . . . . . . . . . 10 (𝑓 = (iEdg‘𝐻) → (𝑓:dom 𝑓1-1→ran 𝑓 ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻)))
1510, 14elab 3319 . . . . . . . . 9 ((iEdg‘𝐻) ∈ {𝑓𝑓:dom 𝑓1-1→ran 𝑓} ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
169, 15sylbb 208 . . . . . . . 8 ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
17163ad2ant3 1077 . . . . . . 7 ((𝐻𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
18 simp2 1055 . . . . . . 7 ((𝐻𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ∧ (iEdg‘𝐻) ∈ 𝑂) → ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
19 f1ssr 6020 . . . . . . 7 (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻) ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
2017, 18, 19syl2anc 691 . . . . . 6 ((𝐻𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
21203exp 1256 . . . . 5 (𝐻𝑊 → (ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})))
227, 21sylbid 229 . . . 4 (𝐻𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2} → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})))
235, 22syl5bi 231 . . 3 (𝐻𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})))
24233imp 1249 . 2 ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
25 eqid 2610 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
26 eqid 2610 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
2725, 26isusgrs 40386 . . 3 (𝐻𝑊 → (𝐻 ∈ USGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
28273ad2ant1 1075 . 2 ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → (𝐻 ∈ USGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
2924, 28mpbird 246 1 ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  {copab 4642  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   USGraph cusgr 40379 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-edga 25793  df-usgr 40381 This theorem is referenced by:  usgrausgrb  40399
 Copyright terms: Public domain W3C validator