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

Theorem cusgrarn 25988
 Description: In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
Assertion
Ref Expression
cusgrarn (𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
Distinct variable groups:   𝑥,𝐸   𝑥,𝑉

Proof of Theorem cusgrarn
Dummy variables 𝑎 𝑏 𝑘 𝑙 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscusgra0 25986 . 2 (𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸))
2 usgraf0 25877 . . 3 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
3 f1f 6014 . . . . . . 7 (𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
4 df-f 5808 . . . . . . . 8 (𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
5 ssel 3562 . . . . . . . . 9 (ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
65adantl 481 . . . . . . . 8 ((𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
74, 6sylbi 206 . . . . . . 7 (𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
83, 7syl 17 . . . . . 6 (𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
98adantr 480 . . . . 5 ((𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
10 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝑒 → (#‘𝑥) = (#‘𝑒))
1110eqeq1d 2612 . . . . . . . . 9 (𝑥 = 𝑒 → ((#‘𝑥) = 2 ↔ (#‘𝑒) = 2))
1211elrab 3331 . . . . . . . 8 (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ (𝑒 ∈ 𝒫 𝑉 ∧ (#‘𝑒) = 2))
13 vex 3176 . . . . . . . . . . 11 𝑒 ∈ V
14 hash2prde 13109 . . . . . . . . . . 11 ((𝑒 ∈ V ∧ (#‘𝑒) = 2) → ∃𝑎𝑏(𝑎𝑏𝑒 = {𝑎, 𝑏}))
1513, 14mpan 702 . . . . . . . . . 10 ((#‘𝑒) = 2 → ∃𝑎𝑏(𝑎𝑏𝑒 = {𝑎, 𝑏}))
16 eleq1 2676 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
17 prex 4836 . . . . . . . . . . . . . . . . 17 {𝑎, 𝑏} ∈ V
1817elpw 4114 . . . . . . . . . . . . . . . 16 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
19 vex 3176 . . . . . . . . . . . . . . . . 17 𝑎 ∈ V
20 vex 3176 . . . . . . . . . . . . . . . . 17 𝑏 ∈ V
2119, 20prss 4291 . . . . . . . . . . . . . . . 16 ((𝑎𝑉𝑏𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉)
2218, 21bitr4i 266 . . . . . . . . . . . . . . 15 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ (𝑎𝑉𝑏𝑉))
23 simprl 790 . . . . . . . . . . . . . . . . . . . 20 ((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑉)
2423anim1i 590 . . . . . . . . . . . . . . . . . . 19 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → (𝑎𝑉𝑎𝑏))
25 eldifsn 4260 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (𝑉 ∖ {𝑏}) ↔ (𝑎𝑉𝑎𝑏))
2624, 25sylibr 223 . . . . . . . . . . . . . . . . . 18 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
27 simplrr 797 . . . . . . . . . . . . . . . . . . 19 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → 𝑏𝑉)
28 sneq 4135 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑏 → {𝑘} = {𝑏})
2928difeq2d 3690 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑏 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑏}))
30 preq2 4213 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑏 → {𝑙, 𝑘} = {𝑙, 𝑏})
3130eleq1d 2672 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑏 → ({𝑙, 𝑘} ∈ ran 𝐸 ↔ {𝑙, 𝑏} ∈ ran 𝐸))
3229, 31raleqbidv 3129 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑏 → (∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
3332rspcv 3278 . . . . . . . . . . . . . . . . . . 19 (𝑏𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
3427, 33syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
35 preq1 4212 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑎 → {𝑙, 𝑏} = {𝑎, 𝑏})
3635eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑎 → ({𝑙, 𝑏} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
3736rspcv 3278 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (𝑉 ∖ {𝑏}) → (∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸 → {𝑎, 𝑏} ∈ ran 𝐸))
3826, 34, 37sylsyld 59 . . . . . . . . . . . . . . . . 17 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → {𝑎, 𝑏} ∈ ran 𝐸))
39 eleq1 2676 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
4039bicomd 212 . . . . . . . . . . . . . . . . . . 19 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
4140adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) → ({𝑎, 𝑏} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
4241adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → ({𝑎, 𝑏} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
4338, 42sylibd 228 . . . . . . . . . . . . . . . 16 (((𝑒 = {𝑎, 𝑏} ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑎𝑏) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
4443exp31 628 . . . . . . . . . . . . . . 15 (𝑒 = {𝑎, 𝑏} → ((𝑎𝑉𝑏𝑉) → (𝑎𝑏 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))))
4522, 44syl5bi 231 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ 𝒫 𝑉 → (𝑎𝑏 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))))
4616, 45sylbid 229 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 𝑉 → (𝑎𝑏 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))))
4746com23 84 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → (𝑎𝑏 → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))))
4847impcom 445 . . . . . . . . . . 11 ((𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸)))
4948exlimivv 1847 . . . . . . . . . 10 (∃𝑎𝑏(𝑎𝑏𝑒 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸)))
5015, 49syl 17 . . . . . . . . 9 ((#‘𝑒) = 2 → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸)))
5150impcom 445 . . . . . . . 8 ((𝑒 ∈ 𝒫 𝑉 ∧ (#‘𝑒) = 2) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
5212, 51sylbi 206 . . . . . . 7 (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸𝑒 ∈ ran 𝐸))
5352com12 32 . . . . . 6 (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑒 ∈ ran 𝐸))
5453adantl 481 . . . . 5 ((𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑒 ∈ ran 𝐸))
559, 54impbid 201 . . . 4 ((𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ ran 𝐸𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
5655eqrdv 2608 . . 3 ((𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
572, 56sylan 487 . 2 ((𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
581, 57syl 17 1 (𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859   ComplUSGrph ccusgra 25947 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-rmo 2904  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-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  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-hash 12980  df-usgra 25862  df-cusgra 25950 This theorem is referenced by:  cusgrafilem1  26007
 Copyright terms: Public domain W3C validator