Theorem usgrislfuspgr 29167

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

Step	Hyp	Ref	Expression
1		usgruspgr 29160	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		usgrislfuspgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3		usgrislfuspgr.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	usgrfs 29137	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		f1f 6738	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
6		2re 12236	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
7	6	leidi 11688	. . . . . . . . . 10 ⊢ 2 ≤ 2
8	7	a1i 11	. . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2)
9		breq2 5106	. . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ 2))
10	8, 9	mpbird 257	. . . . . . . 8 ⊢ ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥))
11	10	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑉 → ((♯‘𝑥) = 2 → 2 ≤ (♯‘𝑥)))
12	11	ss2rabi 4036	. . . . . 6 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
13	12	a1i 11	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
14	5, 13	fssd 6687	. . . 4 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
15	4, 14	syl 17	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
16	1, 15	jca 511	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
17	2, 3	uspgrf 29134	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18		df-f1 6504	. . . . . 6 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ Fun ◡𝐼))
19		fin 6722	. . . . . . . . . . 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 29102	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}
21		feq3 6650	. . . . . . . . . . . 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}))
22	20, 21	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐼:dom 𝐼⟶({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
23	19, 22	sylbb1 237	. . . . . . . . . 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})
24	23	anim1i 615	. . . . . . . . 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 6504	. . . . . . . . 9 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2} ∧ Fun ◡𝐼))
26	24, 25	sylibr 234	. . . . . . . 8 ⊢ (((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ Fun ◡𝐼) → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
27	26	ex 412	. . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (Fun ◡𝐼 → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
28	27	impancom 451	. . . . . 6 ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ Fun ◡𝐼) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
29	18, 28	sylbi 217	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
30	29	imp 406	. . . 4 ⊢ ((𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
31	17, 30	sylan 580	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
32	2, 3	isusgr 29133	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
33	32	adantr 480	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐺 ∈ USGraph ↔ 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
34	31, 33	mpbird 257	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐺 ∈ USGraph)
35	16, 34	impbii 209	1 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585   class class class wbr 5102  ^◡ccnv 5630  dom cdm 5631  Fun wfun 6493  ⟶wf 6495  –1-1→wf1 6496  ‘cfv 6499   ≤ cle 11185  2c2 12217  ♯chash 14271  Vtxcvtx 28976  iEdgciedg 28977  USPGraphcuspgr 29128  USGraphcusgr 29129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-hash 14272  df-uspgr 29130  df-usgr 29131

This theorem is referenced by:  usgr1vr  29235