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 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrawopreg2	⊢ ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (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.

Step	Hyp	Ref	Expression
1		frgrawopreg.a	. . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2		frgrawopreg.b	. . . . . 6 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
3	1, 2	frgrawopreglem1 26571	. . . . 5 ⊢ (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	simprd 478	. . . 4 ⊢ (𝑉 FriendGrph 𝐸 → 𝐵 ∈ V)
5		hash1snb 13068	. . . 4 ⊢ (𝐵 ∈ V → ((#‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣}))
6	4, 5	syl 17	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → ((#‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣}))
7		exsnrex 4168	. . . . 5 ⊢ (∃𝑣 𝐵 = {𝑣} ↔ ∃𝑣 ∈ 𝐵 𝐵 = {𝑣})
8		difss 3699	. . . . . . . 8 ⊢ (𝑉 ∖ 𝐴) ⊆ 𝑉
9	2, 8	eqsstri 3598	. . . . . . 7 ⊢ 𝐵 ⊆ 𝑉
10		ssrexv 3630	. . . . . . 7 ⊢ (𝐵 ⊆ 𝑉 → (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐵 = {𝑣}))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐵 = {𝑣})
12	1, 2	frgrawopreglem4 26574	. . . . . . . 8 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑤 ∈ 𝐴 ∀𝑢 ∈ 𝐵 {𝑤, 𝑢} ∈ ran 𝐸)
13		ralcom 3079	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑢 ∈ 𝐵 {𝑤, 𝑢} ∈ ran 𝐸 ↔ ∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑢} ∈ ran 𝐸)
14		vsnid 4156	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ {𝑣}
15		eleq2 2677	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑣} → (𝑣 ∈ 𝐵 ↔ 𝑣 ∈ {𝑣}))
16	14, 15	mpbiri 247	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑣} → 𝑣 ∈ 𝐵)
17		preq2 4213	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑣 → {𝑤, 𝑢} = {𝑤, 𝑣})
18	17	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑣 → ({𝑤, 𝑢} ∈ ran 𝐸 ↔ {𝑤, 𝑣} ∈ ran 𝐸))
19	18	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (∀𝑤 ∈ 𝐴 {𝑤, 𝑢} ∈ ran 𝐸 ↔ ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
20	19	rspcv 3278	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐵 → (∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
21	16, 20	syl 17	. . . . . . . . . 10 ⊢ (𝐵 = {𝑣} → (∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ ran 𝐸))
22	2	eqeq1i 2615	. . . . . . . . . . . 12 ⊢ (𝐵 = {𝑣} ↔ (𝑉 ∖ 𝐴) = {𝑣})
23		ssrab2 3650	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ⊆ 𝑉
24	1, 23	eqsstri 3598	. . . . . . . . . . . . . 14 ⊢ 𝐴 ⊆ 𝑉
25		dfss4 3820	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ 𝑉 ↔ (𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴)
26		eqcom 2617	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∖ (𝑉 ∖ 𝐴)) = 𝐴 ↔ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
27	25, 26	bitri 263	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ 𝑉 ↔ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴)))
28	24, 27	mpbi 219	. . . . . . . . . . . . 13 ⊢ 𝐴 = (𝑉 ∖ (𝑉 ∖ 𝐴))
29		difeq2 3684	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∖ 𝐴) = {𝑣} → (𝑉 ∖ (𝑉 ∖ 𝐴)) = (𝑉 ∖ {𝑣}))
30	28, 29	syl5eq 2656	. . . . . . . . . . . 12 ⊢ ((𝑉 ∖ 𝐴) = {𝑣} → 𝐴 = (𝑉 ∖ {𝑣}))
31	22, 30	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑣} → 𝐴 = (𝑉 ∖ {𝑣}))
32		prcom 4211	. . . . . . . . . . . . 13 ⊢ {𝑤, 𝑣} = {𝑣, 𝑤}
33	32	eleq1i 2679	. . . . . . . . . . . 12 ⊢ ({𝑤, 𝑣} ∈ ran 𝐸 ↔ {𝑣, 𝑤} ∈ ran 𝐸)
34	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝐵 = {𝑣} → ({𝑤, 𝑣} ∈ ran 𝐸 ↔ {𝑣, 𝑤} ∈ ran 𝐸))
35	31, 34	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝐵 = {𝑣} → (∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ ran 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
36	21, 35	sylibd 228	. . . . . . . . 9 ⊢ (𝐵 = {𝑣} → (∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
37	13, 36	syl5bi 231	. . . . . . . 8 ⊢ (𝐵 = {𝑣} → (∀𝑤 ∈ 𝐴 ∀𝑢 ∈ 𝐵 {𝑤, 𝑢} ∈ ran 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
38	12, 37	syl5com 31	. . . . . . 7 ⊢ (𝑉 FriendGrph 𝐸 → (𝐵 = {𝑣} → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
39	38	reximdv 2999	. . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑣 ∈ 𝑉 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
40	11, 39	syl5com 31	. . . . 5 ⊢ (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
41	7, 40	sylbi 206	. . . 4 ⊢ (∃𝑣 𝐵 = {𝑣} → (𝑉 FriendGrph 𝐸 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
42	41	com12 32	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → (∃𝑣 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
43	6, 42	sylbid 229	. 2 ⊢ (𝑉 FriendGrph 𝐸 → ((#‘𝐵) = 1 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
44	43	imp 444	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

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