Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ausgrusgrb Structured version   Visualization version   GIF version

Theorem ausgrusgrb 40395
 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 ((𝑉𝑋𝐸𝑌) → (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
StepHypRef Expression
1 ausgr.1 . . 3 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}
21isausgr 40394 . 2 ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
3 f1oi 6086 . . . . 5 ( I ↾ 𝐸):𝐸1-1-onto𝐸
4 dff1o5 6059 . . . . . 6 (( I ↾ 𝐸):𝐸1-1-onto𝐸 ↔ (( I ↾ 𝐸):𝐸1-1𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸))
5 f1ss 6019 . . . . . . . . . 10 ((( I ↾ 𝐸):𝐸1-1𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) → ( I ↾ 𝐸):𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
6 dmresi 5376 . . . . . . . . . . . 12 dom ( I ↾ 𝐸) = 𝐸
76eqcomi 2619 . . . . . . . . . . 11 𝐸 = dom ( I ↾ 𝐸)
8 f1eq2 6010 . . . . . . . . . . 11 (𝐸 = dom ( I ↾ 𝐸) → (( I ↾ 𝐸):𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
97, 8ax-mp 5 . . . . . . . . . 10 (( I ↾ 𝐸):𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
105, 9sylib 207 . . . . . . . . 9 ((( I ↾ 𝐸):𝐸1-1𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
1110ex 449 . . . . . . . 8 (( I ↾ 𝐸):𝐸1-1𝐸 → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
1211a1d 25 . . . . . . 7 (( I ↾ 𝐸):𝐸1-1𝐸 → ((𝑉𝑋𝐸𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})))
1312adantr 480 . . . . . 6 ((( I ↾ 𝐸):𝐸1-1𝐸 ∧ ran ( I ↾ 𝐸) = 𝐸) → ((𝑉𝑋𝐸𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})))
144, 13sylbi 206 . . . . 5 (( I ↾ 𝐸):𝐸1-1-onto𝐸 → ((𝑉𝑋𝐸𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})))
153, 14ax-mp 5 . . . 4 ((𝑉𝑋𝐸𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
16 f1f 6014 . . . . . 6 (( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
17 df-f 5808 . . . . . . 7 (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ (( I ↾ 𝐸) Fn dom ( I ↾ 𝐸) ∧ ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
18 rnresi 5398 . . . . . . . . . . 11 ran ( I ↾ 𝐸) = 𝐸
1918sseq1i 3592 . . . . . . . . . 10 (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
2019biimpi 205 . . . . . . . . 9 (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
2120a1d 25 . . . . . . . 8 (ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ((𝑉𝑋𝐸𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2221adantl 481 . . . . . . 7 ((( I ↾ 𝐸) Fn dom ( I ↾ 𝐸) ∧ ran ( I ↾ 𝐸) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) → ((𝑉𝑋𝐸𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2317, 22sylbi 206 . . . . . 6 (( I ↾ 𝐸):dom ( I ↾ 𝐸)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ((𝑉𝑋𝐸𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2416, 23syl 17 . . . . 5 (( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → ((𝑉𝑋𝐸𝑌) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2524com12 32 . . . 4 ((𝑉𝑋𝐸𝑌) → (( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2615, 25impbid 201 . . 3 ((𝑉𝑋𝐸𝑌) → (𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝐸):dom ( I ↾ 𝐸)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
27 resiexg 6994 . . . . 5 (𝐸𝑌 → ( I ↾ 𝐸) ∈ V)
28 opiedgfv 25684 . . . . 5 ((𝑉𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
2927, 28sylan2 490 . . . 4 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
3029dmeqd 5248 . . . 4 ((𝑉𝑋𝐸𝑌) → dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = dom ( I ↾ 𝐸))
31 opvtxfv 25681 . . . . . . 7 ((𝑉𝑋 ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
3227, 31sylan2 490 . . . . . 6 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
3332pweqd 4113 . . . . 5 ((𝑉𝑋𝐸𝑌) → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝒫 𝑉)
3433rabeqdv 3167 . . . 4 ((𝑉𝑋𝐸𝑌) → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
3529, 30, 34f1eq123d 6044 . . 3 ((𝑉𝑋𝐸𝑌) → ((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}))
3626, 35bitr4d 270 . 2 ((𝑉𝑋𝐸𝑌) → (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}))
37 opex 4859 . . . . 5 𝑉, ( I ↾ 𝐸)⟩ ∈ V
38 eqid 2610 . . . . . 6 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
39 eqid 2610 . . . . . 6 (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
4038, 39isusgrs 40386 . . . . 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}))
4137, 40ax-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})
4241bicomi 213 . . 3 ((iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (#‘𝑥) = 2} ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph )
4342a1i 11 . 2 ((𝑉𝑋𝐸𝑌) → ((iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) ∣ (#‘𝑥) = 2} ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ))
442, 36, 433bitrd 293 1 ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸 ↔ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ USGraph ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131   class class class wbr 4583  {copab 4642   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   USGraph cusgr 40379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-vtx 25675  df-iedg 25676  df-usgr 40381 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator