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Theorem frgrawopreg2 26578
 Description: According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
Hypotheses
Ref Expression
frgrawopreg.a 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreg2 ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝐵   𝑤,𝑣,𝑥,𝐴   𝑣,𝐵,𝑤   𝑣,𝑉,𝑤   𝑣,𝐸,𝑤
Allowed substitution hints:   𝐾(𝑤,𝑣)

Proof of Theorem frgrawopreg2
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 frgrawopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2 frgrawopreg.b . . . . . 6 𝐵 = (𝑉𝐴)
31, 2frgrawopreglem1 26571 . . . . 5 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43simprd 478 . . . 4 (𝑉 FriendGrph 𝐸𝐵 ∈ V)
5 hash1snb 13068 . . . 4 (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣}))
64, 5syl 17 . . 3 (𝑉 FriendGrph 𝐸 → ((#‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣}))
7 exsnrex 4168 . . . . 5 (∃𝑣 𝐵 = {𝑣} ↔ ∃𝑣𝐵 𝐵 = {𝑣})
8 difss 3699 . . . . . . . 8 (𝑉𝐴) ⊆ 𝑉
92, 8eqsstri 3598 . . . . . . 7 𝐵𝑉
10 ssrexv 3630 . . . . . . 7 (𝐵𝑉 → (∃𝑣𝐵 𝐵 = {𝑣} → ∃𝑣𝑉 𝐵 = {𝑣}))
119, 10ax-mp 5 . . . . . 6 (∃𝑣𝐵 𝐵 = {𝑣} → ∃𝑣𝑉 𝐵 = {𝑣})
121, 2frgrawopreglem4 26574 . . . . . . . 8 (𝑉 FriendGrph 𝐸 → ∀𝑤𝐴𝑢𝐵 {𝑤, 𝑢} ∈ ran 𝐸)
13 ralcom 3079 . . . . . . . . 9 (∀𝑤𝐴𝑢𝐵 {𝑤, 𝑢} ∈ ran 𝐸 ↔ ∀𝑢𝐵𝑤𝐴 {𝑤, 𝑢} ∈ ran 𝐸)
14 vsnid 4156 . . . . . . . . . . . 12 𝑣 ∈ {𝑣}
15 eleq2 2677 . . . . . . . . . . . 12 (𝐵 = {𝑣} → (𝑣𝐵𝑣 ∈ {𝑣}))
1614, 15mpbiri 247 . . . . . . . . . . 11 (𝐵 = {𝑣} → 𝑣𝐵)
17 preq2 4213 . . . . . . . . . . . . . 14 (𝑢 = 𝑣 → {𝑤, 𝑢} = {𝑤, 𝑣})
1817eleq1d 2672 . . . . . . . . . . . . 13 (𝑢 = 𝑣 → ({𝑤, 𝑢} ∈ ran 𝐸 ↔ {𝑤, 𝑣} ∈ ran 𝐸))
1918ralbidv 2969 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (∀𝑤𝐴 {𝑤, 𝑢} ∈ ran 𝐸 ↔ ∀𝑤𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
2019rspcv 3278 . . . . . . . . . . 11 (𝑣𝐵 → (∀𝑢𝐵𝑤𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
2116, 20syl 17 . . . . . . . . . 10 (𝐵 = {𝑣} → (∀𝑢𝐵𝑤𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
222eqeq1i 2615 . . . . . . . . . . . 12 (𝐵 = {𝑣} ↔ (𝑉𝐴) = {𝑣})
23 ssrab2 3650 . . . . . . . . . . . . . . 15 {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ⊆ 𝑉
241, 23eqsstri 3598 . . . . . . . . . . . . . 14 𝐴𝑉
25 dfss4 3820 . . . . . . . . . . . . . . 15 (𝐴𝑉 ↔ (𝑉 ∖ (𝑉𝐴)) = 𝐴)
26 eqcom 2617 . . . . . . . . . . . . . . 15 ((𝑉 ∖ (𝑉𝐴)) = 𝐴𝐴 = (𝑉 ∖ (𝑉𝐴)))
2725, 26bitri 263 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 = (𝑉 ∖ (𝑉𝐴)))
2824, 27mpbi 219 . . . . . . . . . . . . 13 𝐴 = (𝑉 ∖ (𝑉𝐴))
29 difeq2 3684 . . . . . . . . . . . . 13 ((𝑉𝐴) = {𝑣} → (𝑉 ∖ (𝑉𝐴)) = (𝑉 ∖ {𝑣}))
3028, 29syl5eq 2656 . . . . . . . . . . . 12 ((𝑉𝐴) = {𝑣} → 𝐴 = (𝑉 ∖ {𝑣}))
3122, 30sylbi 206 . . . . . . . . . . 11 (𝐵 = {𝑣} → 𝐴 = (𝑉 ∖ {𝑣}))
32 prcom 4211 . . . . . . . . . . . . 13 {𝑤, 𝑣} = {𝑣, 𝑤}
3332eleq1i 2679 . . . . . . . . . . . 12 ({𝑤, 𝑣} ∈ ran 𝐸 ↔ {𝑣, 𝑤} ∈ ran 𝐸)
3433a1i 11 . . . . . . . . . . 11 (𝐵 = {𝑣} → ({𝑤, 𝑣} ∈ ran 𝐸 ↔ {𝑣, 𝑤} ∈ ran 𝐸))
3531, 34raleqbidv 3129 . . . . . . . . . 10 (𝐵 = {𝑣} → (∀𝑤𝐴 {𝑤, 𝑣} ∈ ran 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
3621, 35sylibd 228 . . . . . . . . 9 (𝐵 = {𝑣} → (∀𝑢𝐵𝑤𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
3713, 36syl5bi 231 . . . . . . . 8 (𝐵 = {𝑣} → (∀𝑤𝐴𝑢𝐵 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
3812, 37syl5com 31 . . . . . . 7 (𝑉 FriendGrph 𝐸 → (𝐵 = {𝑣} → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
3938reximdv 2999 . . . . . 6 (𝑉 FriendGrph 𝐸 → (∃𝑣𝑉 𝐵 = {𝑣} → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
4011, 39syl5com 31 . . . . 5 (∃𝑣𝐵 𝐵 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
417, 40sylbi 206 . . . 4 (∃𝑣 𝐵 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
4241com12 32 . . 3 (𝑉 FriendGrph 𝐸 → (∃𝑣 𝐵 = {𝑣} → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
436, 42sylbid 229 . 2 (𝑉 FriendGrph 𝐸 → ((#‘𝐵) = 1 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
4443imp 444 1 ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1c1 9816  #chash 12979   VDeg cvdg 26420   FriendGrph cfrgra 26515 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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-frgra 26516 This theorem is referenced by:  frgraregorufr0  26579
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