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Theorem usgrexi 40661
 Description: An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
Hypothesis
Ref Expression
usgrexi.p 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}
Assertion
Ref Expression
usgrexi (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph )
Distinct variable groups:   𝑥,𝑃   𝑥,𝑉   𝑥,𝑊

Proof of Theorem usgrexi
StepHypRef Expression
1 f1oi 6086 . . . . . 6 ( I ↾ 𝑃):𝑃1-1-onto𝑃
2 f1of1 6049 . . . . . 6 (( I ↾ 𝑃):𝑃1-1-onto𝑃 → ( I ↾ 𝑃):𝑃1-1𝑃)
31, 2ax-mp 5 . . . . 5 ( I ↾ 𝑃):𝑃1-1𝑃
4 dmresi 5376 . . . . . 6 dom ( I ↾ 𝑃) = 𝑃
5 f1eq2 6010 . . . . . 6 (dom ( I ↾ 𝑃) = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃 ↔ ( I ↾ 𝑃):𝑃1-1𝑃))
64, 5ax-mp 5 . . . . 5 (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃 ↔ ( I ↾ 𝑃):𝑃1-1𝑃)
73, 6mpbir 220 . . . 4 ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃
8 usgrexi.p . . . . . 6 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}
98eqcomi 2619 . . . . 5 {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = 𝑃
10 f1eq3 6011 . . . . 5 ({𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = 𝑃 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃))
119, 10mp1i 13 . . . 4 (𝑉𝑊 → (( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1𝑃))
127, 11mpbiri 247 . . 3 (𝑉𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
13 pwexg 4776 . . . . . . 7 (𝑉𝑊 → 𝒫 𝑉 ∈ V)
148, 13rabexd 4741 . . . . . 6 (𝑉𝑊𝑃 ∈ V)
15 resiexg 6994 . . . . . 6 (𝑃 ∈ V → ( I ↾ 𝑃) ∈ V)
1614, 15syl 17 . . . . 5 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
17 opiedgfv 25684 . . . . 5 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
1816, 17mpdan 699 . . . 4 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
1918dmeqd 5248 . . . 4 (𝑉𝑊 → dom (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = dom ( I ↾ 𝑃))
20 opvtxfv 25681 . . . . . . 7 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
2116, 20mpdan 699 . . . . . 6 (𝑉𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
2221pweqd 4113 . . . . 5 (𝑉𝑊 → 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝒫 𝑉)
2322rabeqdv 3167 . . . 4 (𝑉𝑊 → {𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
2418, 19, 23f1eq123d 6044 . . 3 (𝑉𝑊 → ((iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∣ (#‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
2512, 24mpbird 246 . 2 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∣ (#‘𝑥) = 2})
26 opex 4859 . . 3 𝑉, ( I ↾ 𝑃)⟩ ∈ V
27 eqid 2610 . . . 4 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
28 eqid 2610 . . . 4 (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
2927, 28isusgrs 40386 . . 3 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∣ (#‘𝑥) = 2}))
3026, 29mp1i 13 . 2 (𝑉𝑊 → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ↔ (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩):dom (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)–1-1→{𝑥 ∈ 𝒫 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∣ (#‘𝑥) = 2}))
3125, 30mpbird 246 1 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108  ⟨cop 4131   I cid 4948  dom cdm 5038   ↾ cres 5040  –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:  cusgrexi  40662
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