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Theorem cusconngra 25970
Description: A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
Assertion
Ref Expression
cusconngra (𝑉 ComplUSGrph 𝐸𝑉 ConnGrph 𝐸)

Proof of Theorem cusconngra
Dummy variables 𝑓 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusisusgra 25753 . . 3 (𝑉 ComplUSGrph 𝐸𝑉 USGrph 𝐸)
2 usgrav 25633 . . 3 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
31, 2syl 17 . 2 (𝑉 ComplUSGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4 simp-4l 801 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
5 simpr 475 . . . . . . . . . 10 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) → 𝑘𝑉)
6 eldifi 3693 . . . . . . . . . 10 (𝑛 ∈ (𝑉 ∖ {𝑘}) → 𝑛𝑉)
75, 6anim12i 587 . . . . . . . . 9 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → (𝑘𝑉𝑛𝑉))
87adantr 479 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝑘𝑉𝑛𝑉))
9 usgraf1o 25653 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
109adantl 480 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
11 prcom 4210 . . . . . . . . . . . . . . 15 {𝑛, 𝑘} = {𝑘, 𝑛}
1211eleq1i 2678 . . . . . . . . . . . . . 14 ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑘, 𝑛} ∈ ran 𝐸)
13 f1ocnvfv2 6411 . . . . . . . . . . . . . 14 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑘, 𝑛} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
1412, 13sylan2b 490 . . . . . . . . . . . . 13 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
1514ex 448 . . . . . . . . . . . 12 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
1610, 15syl 17 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
1716adantr 479 . . . . . . . . . 10 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
1817adantr 479 . . . . . . . . 9 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}))
1918imp 443 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛})
20 1pthon2v 25889 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑘𝑉𝑛𝑉) ∧ (𝐸‘(𝐸‘{𝑘, 𝑛})) = {𝑘, 𝑛}) → ∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝)
214, 8, 19, 20syl3anc 1317 . . . . . . 7 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛, 𝑘} ∈ ran 𝐸) → ∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝)
2221ex 448 . . . . . 6 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑘})) → ({𝑛, 𝑘} ∈ ran 𝐸 → ∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
2322ralimdva 2944 . . . . 5 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) ∧ 𝑘𝑉) → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
2423ralimdva 2944 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 → ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
2524expimpd 626 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸) → ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
26 iscusgra 25751 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
27 isconngra1 25967 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
2825, 26, 273imtr4d 281 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸𝑉 ConnGrph 𝐸))
293, 28mpcom 37 1 (𝑉 ComplUSGrph 𝐸𝑉 ConnGrph 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2895  Vcvv 3172  cdif 3536  {csn 4124  {cpr 4126   class class class wbr 4577  ccnv 5027  dom cdm 5028  ran crn 5029  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527   USGrph cusg 25625   ComplUSGrph ccusgra 25713   PathOn cpthon 25798   ConnGrph cconngra 25963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-usgra 25628  df-cusgra 25716  df-wlk 25802  df-trail 25803  df-pth 25804  df-wlkon 25808  df-pthon 25810  df-conngra 25964
This theorem is referenced by: (None)
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