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Theorem usgrislfuspgr 26986
Description: A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
Hypotheses
Ref Expression
usgrislfuspgr.v 𝑉 = (Vtx‘𝐺)
usgrislfuspgr.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
usgrislfuspgr (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐼(𝑥)

Proof of Theorem usgrislfuspgr
StepHypRef Expression
1 usgruspgr 26980 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 usgrislfuspgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
3 usgrislfuspgr.i . . . . 5 𝐼 = (iEdg‘𝐺)
42, 3usgrfs 26959 . . . 4 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5 f1f 6567 . . . . 5 (𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
6 2re 11710 . . . . . . . . . . 11 2 ∈ ℝ
76leidi 11174 . . . . . . . . . 10 2 ≤ 2
87a1i 11 . . . . . . . . 9 ((♯‘𝑥) = 2 → 2 ≤ 2)
9 breq2 5057 . . . . . . . . 9 ((♯‘𝑥) = 2 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ 2))
108, 9mpbird 260 . . . . . . . 8 ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥))
1110a1i 11 . . . . . . 7 (𝑥 ∈ 𝒫 𝑉 → ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥)))
1211ss2rabi 4039 . . . . . 6 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
1312a1i 11 . . . . 5 (𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
145, 13fssd 6520 . . . 4 (𝐼:dom 𝐼1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
154, 14syl 17 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
161, 15jca 515 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
172, 3uspgrf 26956 . . . 4 (𝐺 ∈ USPGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18 df-f1 6350 . . . . . 6 (𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ Fun 𝐼))
19 fin 6551 . . . . . . . . . . 11 (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
20 umgrislfupgrlem 26924 . . . . . . . . . . . 12 ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}
21 feq3 6488 . . . . . . . . . . . 12 (({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} → (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
2220, 21ax-mp 5 . . . . . . . . . . 11 (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
2319, 22sylbb1 240 . . . . . . . . . 10 ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
2423anim1i 617 . . . . . . . . 9 (((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ Fun 𝐼) → (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ Fun 𝐼))
25 df-f1 6350 . . . . . . . . 9 (𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ Fun 𝐼))
2624, 25sylibr 237 . . . . . . . 8 (((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ Fun 𝐼) → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
2726ex 416 . . . . . . 7 ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (Fun 𝐼𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
2827impancom 455 . . . . . 6 ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ Fun 𝐼) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
2918, 28sylbi 220 . . . . 5 (𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
3029imp 410 . . . 4 ((𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
3117, 30sylan 583 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
322, 3isusgr 26955 . . . 4 (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
3332adantr 484 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐺 ∈ USGraph ↔ 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
3431, 33mpbird 260 . 2 ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐺 ∈ USGraph)
3516, 34impbii 212 1 (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {crab 3137  cdif 3916  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   class class class wbr 5053  ccnv 5542  dom cdm 5543  Fun wfun 6339  wf 6341  1-1wf1 6342  cfv 6345  cle 10676  2c2 11691  chash 13697  Vtxcvtx 26798  iEdgciedg 26799  USPGraphcuspgr 26950  USGraphcusgr 26951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-fz 12897  df-hash 13698  df-uspgr 26952  df-usgr 26953
This theorem is referenced by:  usgr1vr  27054
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