Theorem cusconngra 26204

Description: A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Assertion

Ref	Expression
cusconngra	⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 ConnGrph 𝐸)

Proof of Theorem cusconngra

Dummy variables 𝑓 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusisusgra 25987	. . 3 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)
2		usgrav 25867	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3	1, 2	syl 17	. 2 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4		simp-4l 802	. . . . . . . 8 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
5		simpr 476	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) → 𝑘 ∈ 𝑉)
6		eldifi 3694	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑘}) → 𝑛 ∈ 𝑉)
7	5, 6	anim12i 588	. . . . . . . . 9 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
8	7	adantr 480	. . . . . . . 8 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑘 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
9		usgraf1o 25887	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
11		prcom 4211	. . . . . . . . . . . . . . 15 ⊢ {𝑛, 𝑘} = {𝑘, 𝑛}
12	11	eleq1i 2679	. . . . . . . . . . . . . 14 ⊢ ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑘, 𝑛} ∈ ran 𝐸)
13		f1ocnvfv2 6433	. . . . . . . . . . . . . 14 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑘, 𝑛} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
14	12, 13	sylan2b 491	. . . . . . . . . . . . 13 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
15	14	ex 449	. . . . . . . . . . . 12 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
16	10, 15	syl 17	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
18	17	adantr 480	. . . . . . . . 9 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
19	18	imp 444	. . . . . . . 8 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
20		1pthon2v 26123	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑘 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ (𝐸‘(◡𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}) → ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝)
21	4, 8, 19, 20	syl3anc 1318	. . . . . . 7 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝)
22	21	ex 449	. . . . . 6 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
23	22	ralimdva 2945	. . . . 5 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘 ∈ 𝑉) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
24	23	ralimdva 2945	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
25	24	expimpd 627	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
26		iscusgra 25985	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
27		isconngra1 26201	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
28	25, 26, 27	3imtr4d 282	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 → 𝑉 ConnGrph 𝐸))
29	3, 28	mpcom 37	1 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 ConnGrph 𝐸)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   USGrph cusg 25859   ComplUSGrph ccusgra 25947   PathOn cpthon 26032   ConnGrph cconngra 26197

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-rep 4699  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-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  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-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-cusgra 25950  df-wlk 26036  df-trail 26037  df-pth 26038  df-wlkon 26042  df-pthon 26044  df-conngra 26198

This theorem is referenced by: (None)