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Theorem usgruspgrb 25969
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 25966 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
2 edgusgr 25948 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑒) = 2))
32simprd 479 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (#‘𝑒) = 2)
43ralrimiva 2960 . . 3 (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2)
51, 4jca 554 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))
6 edgval 25841 . . . . . 6 (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
76raleqdv 3133 . . . . 5 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2))
8 eqid 2621 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
9 eqid 2621 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
108, 9uspgrf 25942 . . . . . 6 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
11 f1f 6058 . . . . . . . . . 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})
12 frn 6010 . . . . . . . . . 10 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1311, 12syl 17 . . . . . . . . 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 3578 . . . . . . . . . . . . . . 15 ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1514expcom 451 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
16 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝑦 → (#‘𝑒) = (#‘𝑦))
1716eqeq1d 2623 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑦 → ((#‘𝑒) = 2 ↔ (#‘𝑦) = 2))
1817rspcv 3291 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2 → (#‘𝑦) = 2))
19 fveq2 6148 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
2019breq1d 4623 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑦) ≤ 2))
2120elrab 3346 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2))
22 eldifi 3710 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
2322anim1i 591 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
2419eqeq1d 2623 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → ((#‘𝑥) = 2 ↔ (#‘𝑦) = 2))
2524elrab 3346 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
2623, 25sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
2726ex 450 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
2827adantr 481 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
2921, 28sylbi 207 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
3018, 29syl9 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})))
3115, 30syld 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})))
3231com13 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})))
3332imp 445 . . . . . . . . . . 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}))
3433ssrdv 3589 . . . . . . . . . 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})
3534ex 450 . . . . . . . . 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}))
3613, 35mpan9 486 . . . . . . . 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})
37 f1ssr 6064 . . . . . . . 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})
3836, 37syldan 487 . . . . . . 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})
3938ex 450 . . . . . 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}))
4010, 39syl 17 . . . . 5 (𝐺 ∈ USPGraph → (∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
417, 40sylbid 230 . . . 4 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
4241imp 445 . . 3 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
438, 9isusgrs 25944 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
4443adantr 481 . . 3 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
4542, 44mpbird 247 . 2 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → 𝐺 ∈ USGraph )
465, 45impbii 199 1 (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  dom cdm 5074  ran crn 5075  wf 5843  1-1wf1 5844  cfv 5847  cle 10019  2c2 11014  #chash 13057  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   USPGraph cuspgr 25936   USGraph cusgr 25937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-edg 25840  df-uspgr 25938  df-usgr 25939
This theorem is referenced by:  usgr1e  26030
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