Theorem ausgrusgrb 29611

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 29610	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
3		f1oi 6860	. . . . 5 ⊢ ( I ↾ 𝐸):𝐸–1-1-onto→𝐸
4		dff1o5 6831	. . . . . 6 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 ↔ (( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5		f1ss 6782	. . . . . . . . . 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 6052	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐸) = 𝐸
7	6	eqcomi 2771	. . . . . . . . . . 11 ⊢ 𝐸 = dom ( I ↾ 𝐸)
8		f1eq2 6771	. . . . . . . . . . 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 221	. . . . . . . . 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 418	. . . . . . . 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 26	. . . . . . 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 486	. . . . . 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 220	. . . . 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 6541	. . . . . 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 6075	. . . . . . . . 9 ⊢ ran ( I ↾ 𝐸) = 𝐸
18	17	sseq1i 3962	. . . . . . . 8 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
19	18	biimpi 219	. . . . . . 7 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
20	19	a1d 26	. . . . . 6 ⊢ (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
21	16, 20	simplbiim 514	. . . . 5 ⊢ (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
22		f1f 6775	. . . . 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 34	. . . 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 215	. . 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 7912	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ( I ↾ 𝐸) ∈ V)
26		opiedgfv 29450	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
27	25, 26	sylan2 605	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
28	27	dmeqd 5893	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
29		opvtxfv 29447	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
30	25, 29	sylan2 605	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
31	30	pweqd 4577	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
32	31	rabeqdv 3429	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
33	27, 28, 32	f1eq123d 6813	. . 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 285	. 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 5443	. . . . 5 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
36		eqid 2762	. . . . . 6 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
37		eqid 2762	. . . . . 6 ⊢ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
38	36, 37	isusgrs 29602	. . . . 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 227	. . 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 308	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  ⟨cop 4593   class class class wbr 5107  {copab 5171   I cid 5553  dom cdm 5659  ran crn 5660   ↾ cres 5661   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  2c2 12322  ♯chash 14396  Vtxcvtx 29439  iEdgciedg 29440  USGraphcusgr 29595

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-hash 14397  df-vtx 29441  df-iedg 29442  df-usgr 29597

This theorem is used by: (None)