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Theorem usgruspgrb 40411
 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 40408 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
2 edgusgr 40391 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑒) = 2))
32simprd 478 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (#‘𝑒) = 2)
43ralrimiva 2949 . . 3 (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2)
51, 4jca 553 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))
6 edgaval 25794 . . . . . 6 (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
76raleqdv 3121 . . . . 5 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2))
8 eqid 2610 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
9 eqid 2610 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
108, 9uspgrf 40384 . . . . . 6 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
11 f1f 6014 . . . . . . . . . 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 5966 . . . . . . . . . 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 3563 . . . . . . . . . . . . . . 15 ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
1514expcom 450 . . . . . . . . . . . . . 14 (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
16 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝑦 → (#‘𝑒) = (#‘𝑦))
1716eqeq1d 2612 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑦 → ((#‘𝑒) = 2 ↔ (#‘𝑦) = 2))
1817rspcv 3278 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2 → (#‘𝑦) = 2))
19 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
2019breq1d 4593 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑦) ≤ 2))
2120elrab 3331 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2))
22 eldifi 3694 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
2322anim1i 590 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
2419eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → ((#‘𝑥) = 2 ↔ (#‘𝑦) = 2))
2524elrab 3331 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
2623, 25sylibr 223 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
2726ex 449 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
2827adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
2921, 28sylbi 206 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
3018, 29syl9 75 . . . . . . . . . . . . . 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 46 . . . . . . . . . . . . 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 86 . . . . . . . . . . . 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 444 . . . . . . . . . . 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 3574 . . . . . . . . . 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 449 . . . . . . . . 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 485 . . . . . . . 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 6020 . . . . . . . 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 486 . . . . . . 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 449 . . . . . 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 229 . . . 4 (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
4241imp 444 . . 3 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
438, 9isusgrs 40386 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
4443adantr 480 . . 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 246 . 2 ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → 𝐺 ∈ USGraph )
465, 45impbii 198 1 (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   USPGraph cuspgr 40378   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-edga 25793  df-uspgr 40380  df-usgr 40381 This theorem is referenced by:  usgr1e  40471
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