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Theorem ausgrusgri 25956
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 6158 . . . . 5 (Vtx‘𝐻) ∈ V
2 fvex 6158 . . . . 5 (Edg‘𝐻) ∈ V
3 ausgr.1 . . . . . 6 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}
43isausgr 25952 . . . . 5 (((Vtx‘𝐻) ∈ V ∧ (Edg‘𝐻) ∈ V) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
51, 2, 4mp2an 707 . . . 4 ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})
6 edgval 25841 . . . . . 6 (𝐻𝑊 → (Edg‘𝐻) = ran (iEdg‘𝐻))
76sseq1d 3611 . . . . 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 2690 . . . . . . . . 9 ((iEdg‘𝐻) ∈ 𝑂 ↔ (iEdg‘𝐻) ∈ {𝑓𝑓:dom 𝑓1-1→ran 𝑓})
10 fvex 6158 . . . . . . . . . 10 (iEdg‘𝐻) ∈ V
11 id 22 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → 𝑓 = (iEdg‘𝐻))
12 dmeq 5284 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → dom 𝑓 = dom (iEdg‘𝐻))
13 rneq 5311 . . . . . . . . . . 11 (𝑓 = (iEdg‘𝐻) → ran 𝑓 = ran (iEdg‘𝐻))
1411, 12, 13f1eq123d 6088 . . . . . . . . . 10 (𝑓 = (iEdg‘𝐻) → (𝑓:dom 𝑓1-1→ran 𝑓 ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻)))
1510, 14elab 3333 . . . . . . . . 9 ((iEdg‘𝐻) ∈ {𝑓𝑓:dom 𝑓1-1→ran 𝑓} ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
169, 15sylbb 209 . . . . . . . 8 ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
17163ad2ant3 1082 . . . . . . 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 1060 . . . . . . 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 6064 . . . . . . 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 692 . . . . . 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 1261 . . . . 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 230 . . . 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 232 . . 3 (𝐻𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2})))
24233imp 1254 . 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 2621 . . . 4 (Vtx‘𝐻) = (Vtx‘𝐻)
26 eqid 2621 . . . 4 (iEdg‘𝐻) = (iEdg‘𝐻)
2725, 26isusgrs 25944 . . 3 (𝐻𝑊 → (𝐻 ∈ USGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ (#‘𝑥) = 2}))
28273ad2ant1 1080 . 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 247 1 ((𝐻𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   class class class wbr 4613  {copab 4672  dom cdm 5074  ran crn 5075  1-1wf1 5844  cfv 5847  2c2 11014  #chash 13057  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   USGraph cusgr 25937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-edg 25840  df-usgr 25939
This theorem is referenced by:  usgrausgrb  25957
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