Theorem ausgrusgrb 29368

Description: The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)

Hypothesis

Ref	Expression
ausgr.1	⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}

Assertion

Ref	Expression
ausgrusgrb	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

Distinct variable groups:   𝑣,𝑒,𝑥,𝐸   𝑒,𝑉,𝑣,𝑥   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐺(𝑥,𝑣,𝑒)   𝑋(𝑣,𝑒)   𝑌(𝑣,𝑒)

Proof of Theorem ausgrusgrb

Step	Hyp	Ref	Expression
1		ausgr.1	. . 3 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2}}
2	1	isausgr 29367	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
3		f1oi 6847	. . . . 5 ⊢ ( I ↾ 𝐸):𝐸–1-1-onto→𝐸
4		dff1o5 6818	. . . . . 6 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 ↔ (( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5		f1ss 6769	. . . . . . . . . 10 ⊢ ((( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) → ( I ↾ 𝐸):𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
6		dmresi 6043	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐸) = 𝐸
7	6	eqcomi 2773	. . . . . . . . . . 11 ⊢ 𝐸 = dom ( I ↾ 𝐸)
8		f1eq2 6758	. . . . . . . . . . 11 ⊢ (𝐸 = dom ( I ↾ 𝐸) → (( I ↾ 𝐸):𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ (( I ↾ 𝐸):𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
10	5, 9	sylib 220	. . . . . . . . 9 ⊢ ((( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
11	10	ex 416	. . . . . . . 8 ⊢ (( I ↾ 𝐸):𝐸–1-1→𝐸 → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
12	11	a1d 25	. . . . . . 7 ⊢ (( I ↾ 𝐸):𝐸–1-1→𝐸 → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})))
13	12	adantr 484	. . . . . 6 ⊢ ((( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})))
14	4, 13	sylbi 219	. . . . 5 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})))
15	3, 14	ax-mp 5	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
16		df-f 6527	. . . . . 6 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ (( I ↾ 𝐸) Fn dom ( I ↾ 𝐸) ∧ ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
17		rnresi 6066	. . . . . . . . 9 ⊢ ran ( I ↾ 𝐸) = 𝐸
18	17	sseq1i 3966	. . . . . . . 8 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
19	18	biimpi 218	. . . . . . 7 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
20	19	a1d 25	. . . . . 6 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
21	16, 20	simplbiim 512	. . . . 5 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
22		f1f 6762	. . . . 5 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
23	21, 22	syl11 33	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
24	15, 23	impbid 214	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
25		resiexg 7895	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ( I ↾ 𝐸) ∈ V)
26		opiedgfv 29210	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
27	25, 26	sylan2 602	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
28	27	dmeqd 5883	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
29		opvtxfv 29207	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
30	25, 29	sylan2 602	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
31	30	pweqd 4574	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
32	31	rabeqdv 3431	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
33	27, 28, 32	f1eq123d 6800	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
34	24, 33	bitr4d 284	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2}))
35		opex 5433	. . . . 5 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
36		eqid 2764	. . . . . 6 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
37		eqid 2764	. . . . . 6 ⊢ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
38	36, 37	isusgrs 29359	. . . . 5 ⊢ (⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2}))
39	35, 38	ax-mp 5	. . . 4 ⊢ (⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2})
40	39	bicomi 226	. . 3 ⊢ ((iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph)
41	40	a1i 11	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))
42	2, 34, 41	3bitrd 307	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {crab 3416  Vcvv 3456   ⊆ wss 3906  𝒫 cpw 4557  ⟨cop 4590   class class class wbr 5102  {copab 5164   I cid 5543  dom cdm 5649  ran crn 5650   ↾ cres 5651   Fn wfn 6518  ⟶wf 6519  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523  2c2 12274  ♯chash 14345  Vtxcvtx 29199  iEdgciedg 29200  USGraphcusgr 29352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-n0 12484  df-z 12571  df-uz 12842  df-fz 13515  df-hash 14346  df-vtx 29201  df-iedg 29202  df-usgr 29354

This theorem is referenced by: (None)