Theorem ausgrusgrb 29143

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 29142	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
3		f1oi 6801	. . . . 5 ⊢ ( I ↾ 𝐸):𝐸–1-1-onto→𝐸
4		dff1o5 6772	. . . . . 6 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 ↔ (( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5		f1ss 6724	. . . . . . . . . 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 6000	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐸) = 𝐸
7	6	eqcomi 2740	. . . . . . . . . . 11 ⊢ 𝐸 = dom ( I ↾ 𝐸)
8		f1eq2 6715	. . . . . . . . . . 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 218	. . . . . . . . 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 412	. . . . . . . 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 480	. . . . . 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 217	. . . . 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 6485	. . . . . 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 6023	. . . . . . . . 9 ⊢ ran ( I ↾ 𝐸) = 𝐸
18	17	sseq1i 3958	. . . . . . . 8 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
19	18	biimpi 216	. . . . . . 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 504	. . . . 5 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
22		f1f 6719	. . . . 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 212	. . 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 7842	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ( I ↾ 𝐸) ∈ V)
26		opiedgfv 28985	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
27	25, 26	sylan2 593	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
28	27	dmeqd 5844	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
29		opvtxfv 28982	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
30	25, 29	sylan2 593	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
31	30	pweqd 4564	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
32	31	rabeqdv 3410	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
33	27, 28, 32	f1eq123d 6755	. . 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 282	. 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 5402	. . . . 5 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
36		eqid 2731	. . . . . 6 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
37		eqid 2731	. . . . . 6 ⊢ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
38	36, 37	isusgrs 29134	. . . . 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 224	. . 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 305	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  ⟨cop 4579   class class class wbr 5089  {copab 5151   I cid 5508  dom cdm 5614  ran crn 5615   ↾ cres 5616   Fn wfn 6476  ⟶wf 6477  –1-1→wf1 6478  –1-1-onto→wf1o 6480  ‘cfv 6481  2c2 12180  ♯chash 14237  Vtxcvtx 28974  iEdgciedg 28975  USGraphcusgr 29127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-hash 14238  df-vtx 28976  df-iedg 28977  df-usgr 29129

This theorem is referenced by: (None)