Theorem ausgrusgrb 26950

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 26949	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
3		f1oi 6652	. . . . 5 ⊢ ( I ↾ 𝐸):𝐸–1-1-onto→𝐸
4		dff1o5 6624	. . . . . 6 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 ↔ (( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5		f1ss 6580	. . . . . . . . . 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 5921	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐸) = 𝐸
7	6	eqcomi 2830	. . . . . . . . . . 11 ⊢ 𝐸 = dom ( I ↾ 𝐸)
8		f1eq2 6571	. . . . . . . . . . 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 415	. . . . . . . 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 483	. . . . . 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 6359	. . . . . 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 5943	. . . . . . . . 9 ⊢ ran ( I ↾ 𝐸) = 𝐸
18	17	sseq1i 3995	. . . . . . . 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 507	. . . . 5 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
22		f1f 6575	. . . . 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 7619	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ( I ↾ 𝐸) ∈ V)
26		opiedgfv 26792	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
27	25, 26	sylan2 594	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
28	27	dmeqd 5774	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
29		opvtxfv 26789	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
30	25, 29	sylan2 594	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
31	30	pweqd 4558	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
32	31	rabeqdv 3484	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
33	27, 28, 32	f1eq123d 6608	. . 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 5356	. . . . 5 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
36		eqid 2821	. . . . . 6 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
37		eqid 2821	. . . . . 6 ⊢ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
38	36, 37	isusgrs 26941	. . . . 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 398   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539  ⟨cop 4573   class class class wbr 5066  {copab 5128   I cid 5459  dom cdm 5555  ran crn 5556   ↾ cres 5557   Fn wfn 6350  ⟶wf 6351  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  2c2 11693  ♯chash 13691  Vtxcvtx 26781  iEdgciedg 26782  USGraphcusgr 26934

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-vtx 26783  df-iedg 26784  df-usgr 26936

This theorem is referenced by: (None)