Theorem usgrislfuspgr 29278

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 29271	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		usgrislfuspgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3		usgrislfuspgr.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	usgrfs 29248	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5		f1f 6740	. . . . 5 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
6		2re 12233	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
7	6	leidi 11685	. . . . . . . . . 10 ⊢ 2 ≤ 2
8	7	a1i 11	. . . . . . . . 9 ⊢ ((♯‘𝑥) = 2 → 2 ≤ 2)
9		breq2 5104	. . . . . . . . 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 4030	. . . . . 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 6689	. . . 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 29245	. . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18		df-f1 6507	. . . . . 6 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐼:dom 𝐼⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ Fun ◡𝐼))
19		fin 6724	. . . . . . . . . . 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 29213	. . . . . . . . . . . 12 ⊢ ({𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∩ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}
21		feq3 6652	. . . . . . . . . . . 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 616	. . . . . . . . 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 6507	. . . . . . . . 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 581	. . 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 29244	. . . 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 1542   ∈ wcel 2114  {crab 3401   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100  ^◡ccnv 5633  dom cdm 5634  Fun wfun 6496  ⟶wf 6498  –1-1→wf1 6499  ‘cfv 6502   ≤ cle 11181  2c2 12214  ♯chash 14267  Vtxcvtx 29087  iEdgciedg 29088  USPGraphcuspgr 29239  USGraphcusgr 29240

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268  df-uspgr 29241  df-usgr 29242

This theorem is referenced by:  usgr1vr  29346