Theorem ausgrusgrb 29256

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 29255	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))
3		f1oi 6822	. . . . 5 ⊢ ( I ↾ 𝐸):𝐸–1-1-onto→𝐸
4		dff1o5 6793	. . . . . 6 ⊢ (( I ↾ 𝐸):𝐸–1-1-onto→𝐸 ↔ (( I ↾ 𝐸):𝐸–1-1→𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5		f1ss 6745	. . . . . . . . . 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 6021	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐸) = 𝐸
7	6	eqcomi 2746	. . . . . . . . . . 11 ⊢ 𝐸 = dom ( I ↾ 𝐸)
8		f1eq2 6736	. . . . . . . . . . 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 6506	. . . . . 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 6044	. . . . . . . . 9 ⊢ ran ( I ↾ 𝐸) = 𝐸
18	17	sseq1i 3964	. . . . . . . 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 6740	. . . . 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 7866	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → ( I ↾ 𝐸) ∈ V)
26		opiedgfv 29098	. . . . 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 5864	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
29		opvtxfv 29095	. . . . . . 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 4573	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
32	31	rabeqdv 3416	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
33	27, 28, 32	f1eq123d 6776	. . 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 5421	. . . . 5 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
36		eqid 2737	. . . . . 6 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
37		eqid 2737	. . . . . 6 ⊢ (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
38	36, 37	isusgrs 29247	. . . . 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 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ⟨cop 4588   class class class wbr 5100  {copab 5162   I cid 5528  dom cdm 5634  ran crn 5635   ↾ cres 5636   Fn wfn 6497  ⟶wf 6498  –1-1→wf1 6499  –1-1-onto→wf1o 6501  ‘cfv 6502  2c2 12214  ♯chash 14267  Vtxcvtx 29087  iEdgciedg 29088  USGraphcusgr 29240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-vtx 29089  df-iedg 29090  df-usgr 29242

This theorem is referenced by: (None)