Theorem ausgrusgrien 16395

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 ≈ 2o}}
ausgrusgri.1	⊢ 𝑂 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓}

Assertion

Ref	Expression
ausgrusgrien	⊢ ((𝐻 ∈ 𝑊 ∧ (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 ausgrusgrien

Step	Hyp	Ref	Expression
1		vtxex 16242	. . . . 5 ⊢ (𝐻 ∈ 𝑊 → (Vtx‘𝐻) ∈ V)
2		edgvalg 16283	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → (Edg‘𝐻) = ran (iEdg‘𝐻))
3		iedgex 16243	. . . . . . 7 ⊢ (𝐻 ∈ 𝑊 → (iEdg‘𝐻) ∈ V)
4		rnexg 5045	. . . . . . 7 ⊢ ((iEdg‘𝐻) ∈ V → ran (iEdg‘𝐻) ∈ V)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → ran (iEdg‘𝐻) ∈ V)
6	2, 5	eqeltrd 2315	. . . . 5 ⊢ (𝐻 ∈ 𝑊 → (Edg‘𝐻) ∈ V)
7		ausgr.1	. . . . . 6 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
8	7	isausgren 16391	. . . . 5 ⊢ (((Vtx‘𝐻) ∈ V ∧ (Edg‘𝐻) ∈ V) → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}))
9	1, 6, 8	syl2anc 415	. . . 4 ⊢ (𝐻 ∈ 𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) ↔ (Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}))
10	2	sseq1d 3277	. . . . 5 ⊢ (𝐻 ∈ 𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} ↔ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}))
11		ausgrusgri.1	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓}
12	11	eleq2i 2305	. . . . . . . . . 10 ⊢ ((iEdg‘𝐻) ∈ 𝑂 ↔ (iEdg‘𝐻) ∈ {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓})
13	12	biimpi 120	. . . . . . . . 9 ⊢ ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻) ∈ {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓})
14		id 19	. . . . . . . . . . 11 ⊢ (𝑓 = (iEdg‘𝐻) → 𝑓 = (iEdg‘𝐻))
15		dmeq 4979	. . . . . . . . . . 11 ⊢ (𝑓 = (iEdg‘𝐻) → dom 𝑓 = dom (iEdg‘𝐻))
16		rneq 5007	. . . . . . . . . . 11 ⊢ (𝑓 = (iEdg‘𝐻) → ran 𝑓 = ran (iEdg‘𝐻))
17	14, 15, 16	f1eq123d 5629	. . . . . . . . . 10 ⊢ (𝑓 = (iEdg‘𝐻) → (𝑓:dom 𝑓–1-1→ran 𝑓 ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻)))
18	17	elabg 2972	. . . . . . . . 9 ⊢ ((iEdg‘𝐻) ∈ 𝑂 → ((iEdg‘𝐻) ∈ {𝑓 ∣ 𝑓:dom 𝑓–1-1→ran 𝑓} ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻)))
19	13, 18	mpbid 147	. . . . . . . 8 ⊢ ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
20	19	3ad2ant3 1051	. . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻))
21		simp2 1029	. . . . . . 7 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} ∧ (iEdg‘𝐻) ∈ 𝑂) → ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})
22		f1ssr 5603	. . . . . . 7 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→ran (iEdg‘𝐻) ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})
23	20, 21, 22	syl2anc 415	. . . . . 6 ⊢ ((𝐻 ∈ 𝑊 ∧ ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})
24	23	3exp 1233	. . . . 5 ⊢ (𝐻 ∈ 𝑊 → (ran (iEdg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})))
25	10, 24	sylbid 150	. . . 4 ⊢ (𝐻 ∈ 𝑊 → ((Edg‘𝐻) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o} → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})))
26	9, 25	sylbid 150	. . 3 ⊢ (𝐻 ∈ 𝑊 → ((Vtx‘𝐻)𝐺(Edg‘𝐻) → ((iEdg‘𝐻) ∈ 𝑂 → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})))
27	26	3imp 1224	. 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o})
28		eqid 2238	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
29		eqid 2238	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
30	28, 29	isusgren 16382	. . 3 ⊢ (𝐻 ∈ 𝑊 → (𝐻 ∈ USGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}))
31	30	3ad2ant1 1049	. 2 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → (𝐻 ∈ USGraph ↔ (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐻) ∣ 𝑥 ≈ 2o}))
32	27, 31	mpbird 167	1 ⊢ ((𝐻 ∈ 𝑊 ∧ (Vtx‘𝐻)𝐺(Edg‘𝐻) ∧ (iEdg‘𝐻) ∈ 𝑂) → 𝐻 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4128  {copab 4189  dom cdm 4772  ran crn 4773  –1-1→wf1 5372  ‘cfv 5375  2_oc2o 6675   ≈ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  Edgcedg 16281  USGraphcusgr 16378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-usgren 16380

This theorem is referenced by:  usgrausgrben  16396