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Theorem usgruspgrb 29215
Description: A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
Assertion
Ref Expression
usgruspgrb (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
Distinct variable group:   𝑒,𝐺

Proof of Theorem usgruspgrb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgruspgr 29212 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 edgusgr 29192 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑒) = 2))
32simprd 495 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (♯‘𝑒) = 2)
43ralrimiva 3144 . . 3 (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2)
51, 4jca 511 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
6 edgval 29081 . . . . . . 7 (Edg‘𝐺) = ran (iEdg‘𝐺)
76a1i 11 . . . . . 6 (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
87raleqdv 3324 . . . . 5 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2))
9 eqid 2735 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
10 eqid 2735 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
119, 10uspgrf 29186 . . . . . 6 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
12 f1f 6805 . . . . . . . . . 10 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1312frnd 6745 . . . . . . . . 9 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
14 ssel2 3990 . . . . . . . . . . . . . . 15 ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
1514expcom 413 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
16 fveqeq2 6916 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑦 → ((♯‘𝑒) = 2 ↔ (♯‘𝑦) = 2))
1716rspcv 3618 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (♯‘𝑦) = 2))
18 fveq2 6907 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
1918breq1d 5158 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑦) ≤ 2))
2019elrab 3695 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2))
21 eldifi 4141 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
2221anim1i 615 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
23 fveqeq2 6916 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → ((♯‘𝑥) = 2 ↔ (♯‘𝑦) = 2))
2423elrab 3695 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
2522, 24sylibr 234 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
2625ex 412 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
2726adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
2820, 27sylbi 217 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
2917, 28syl9 77 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (iEdg‘𝐺) → (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
3015, 29syld 47 . . . . . . . . . . . . 13 (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
3130com13 88 . . . . . . . . . . . 12 (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
3231imp 406 . . . . . . . . . . 11 ((∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
3332ssrdv 4001 . . . . . . . . . 10 ((∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
3433ex 412 . . . . . . . . 9 (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
3513, 34mpan9 506 . . . . . . . 8 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
36 f1ssr 6811 . . . . . . . 8 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
3735, 36syldan 591 . . . . . . 7 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
3837ex 412 . . . . . 6 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
3911, 38syl 17 . . . . 5 (𝐺 ∈ USPGraph → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
408, 39sylbid 240 . . . 4 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
4140imp 406 . . 3 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
429, 10isusgrs 29188 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
4342adantr 480 . . 3 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
4441, 43mpbird 257 . 2 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → 𝐺 ∈ USGraph)
455, 44impbii 209 1 (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ran crn 5690  1-1wf1 6560  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  USPGraphcuspgr 29180  USGraphcusgr 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-uspgr 29182  df-usgr 29183
This theorem is referenced by:  usgr1e  29277
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