MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgrasscusgra Structured version   Visualization version   GIF version

Theorem usgrasscusgra 25777
Description: An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Assertion
Ref Expression
usgrasscusgra ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
Distinct variable groups:   𝑒,𝐸   𝑒,𝐹,𝑓   𝑒,𝑉
Allowed substitution hints:   𝐸(𝑓)   𝑉(𝑓)

Proof of Theorem usgrasscusgra
Dummy variables 𝑎 𝑏 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrarnedg 25679 . . . 4 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}))
2 iscusgra0 25752 . . . . . . 7 (𝑉 ComplUSGrph 𝐹 → (𝑉 USGrph 𝐹 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹))
3 simplrr 796 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → 𝑏𝑉)
4 sneq 4134 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → {𝑘} = {𝑏})
54difeq2d 3689 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑏}))
6 preq2 4212 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → {𝑛, 𝑘} = {𝑛, 𝑏})
76eleq1d 2671 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → ({𝑛, 𝑘} ∈ ran 𝐹 ↔ {𝑛, 𝑏} ∈ ran 𝐹))
85, 7raleqbidv 3128 . . . . . . . . . . . . 13 (𝑘 = 𝑏 → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
98rspcv 3277 . . . . . . . . . . . 12 (𝑏𝑉 → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
103, 9syl 17 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹))
11 simplrl 795 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑎𝑉)
12 velsn 4140 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ {𝑏} ↔ 𝑎 = 𝑏)
13 nne 2785 . . . . . . . . . . . . . . . . . . 19 𝑎𝑏𝑎 = 𝑏)
1412, 13bitr4i 265 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ {𝑏} ↔ ¬ 𝑎𝑏)
1514biimpi 204 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ {𝑏} → ¬ 𝑎𝑏)
1615a1i 11 . . . . . . . . . . . . . . . 16 ((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎 ∈ {𝑏} → ¬ 𝑎𝑏))
1716con2d 127 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑏 → ¬ 𝑎 ∈ {𝑏}))
1817imp 443 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → ¬ 𝑎 ∈ {𝑏})
1911, 18eldifd 3550 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
2019adantrr 748 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
21 preq1 4211 . . . . . . . . . . . . . 14 (𝑛 = 𝑎 → {𝑛, 𝑏} = {𝑎, 𝑏})
2221eleq1d 2671 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → ({𝑛, 𝑏} ∈ ran 𝐹 ↔ {𝑎, 𝑏} ∈ ran 𝐹))
2322rspcv 3277 . . . . . . . . . . . 12 (𝑎 ∈ (𝑉 ∖ {𝑏}) → (∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
2420, 23syl 17 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑛 ∈ (𝑉 ∖ {𝑏}){𝑛, 𝑏} ∈ ran 𝐹 → {𝑎, 𝑏} ∈ ran 𝐹))
25 eleq1 2675 . . . . . . . . . . . . . 14 ({𝑎, 𝑏} = 𝑒 → ({𝑎, 𝑏} ∈ ran 𝐹𝑒 ∈ ran 𝐹))
2625eqcoms 2617 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐹𝑒 ∈ ran 𝐹))
27 equid 1925 . . . . . . . . . . . . . 14 𝑒 = 𝑒
28 equequ2 1939 . . . . . . . . . . . . . . 15 (𝑓 = 𝑒 → (𝑒 = 𝑓𝑒 = 𝑒))
2928rspcev 3281 . . . . . . . . . . . . . 14 ((𝑒 ∈ ran 𝐹𝑒 = 𝑒) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
3027, 29mpan2 702 . . . . . . . . . . . . 13 (𝑒 ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
3126, 30syl6bi 241 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3231ad2antll 760 . . . . . . . . . . 11 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3310, 24, 323syld 57 . . . . . . . . . 10 (((𝑉 USGrph 𝐹 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑒 = {𝑎, 𝑏})) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
3433exp31 627 . . . . . . . . 9 (𝑉 USGrph 𝐹 → ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))))
3534com24 92 . . . . . . . 8 (𝑉 USGrph 𝐹 → (∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹 → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))))
3635imp 443 . . . . . . 7 ((𝑉 USGrph 𝐹 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐹) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
372, 36syl 17 . . . . . 6 (𝑉 ComplUSGrph 𝐹 → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → ((𝑎𝑉𝑏𝑉) → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
3837com13 85 . . . . 5 ((𝑎𝑉𝑏𝑉) → ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓)))
3938rexlimivv 3017 . . . 4 (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
401, 39syl 17 . . 3 ((𝑉 USGrph 𝐸𝑒 ∈ ran 𝐸) → (𝑉 ComplUSGrph 𝐹 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
4140impancom 454 . 2 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (𝑒 ∈ ran 𝐸 → ∃𝑓 ∈ ran 𝐹 𝑒 = 𝑓))
4241ralrimiv 2947 1 ((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  {csn 4124  {cpr 4126   class class class wbr 4577  ran crn 5029   USGrph cusg 25625   ComplUSGrph ccusgra 25713
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-rmo 2903  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-2o 7425  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  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-hash 12935  df-usgra 25628  df-cusgra 25716
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator