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Theorem frgrawopreglem4 26340
Description: Lemma 4 for frgrawopreg 26342. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreglem4 (𝑉 FriendGrph 𝐸 → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ ran 𝐸)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝐴,𝑏   𝑥,𝑎,𝑏,𝐸   𝑉,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑥,𝑎,𝑏)   𝐾(𝑎,𝑏)

Proof of Theorem frgrawopreglem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frgrawopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
2 frgrawopreg.b . . . 4 𝐵 = (𝑉𝐴)
31, 2frgrawopreglem3 26339 . . 3 ((𝑎𝐴𝑏𝐵) → ((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏))
4 frgrancvvdgeq 26336 . . . 4 (𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)))
5 elrabi 3327 . . . . . . . . 9 (𝑎 ∈ {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} → 𝑎𝑉)
65, 1eleq2s 2705 . . . . . . . 8 (𝑎𝐴𝑎𝑉)
7 sneq 4134 . . . . . . . . . . 11 (𝑥 = 𝑎 → {𝑥} = {𝑎})
87difeq2d 3689 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑎}))
9 oveq2 6535 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝑎))
10 neleq2 2888 . . . . . . . . . . . 12 ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → (𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↔ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎)))
119, 10syl 17 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↔ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎)))
12 fveq2 6088 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑎))
1312eqeq1d 2611 . . . . . . . . . . 11 (𝑥 = 𝑎 → (((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦) ↔ ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦)))
1411, 13imbi12d 332 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) ↔ (𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦))))
158, 14raleqbidv 3128 . . . . . . . . 9 (𝑥 = 𝑎 → (∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) ↔ ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦))))
1615rspcv 3277 . . . . . . . 8 (𝑎𝑉 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦))))
176, 16syl 17 . . . . . . 7 (𝑎𝐴 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦))))
1817adantr 479 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦))))
192eleq2i 2679 . . . . . . . . . 10 (𝑏𝐵𝑏 ∈ (𝑉𝐴))
20 eldif 3549 . . . . . . . . . 10 (𝑏 ∈ (𝑉𝐴) ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
2119, 20bitri 262 . . . . . . . . 9 (𝑏𝐵 ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
22 simpll 785 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏𝑉)
23 eleq1a 2682 . . . . . . . . . . . . . . 15 (𝑎𝐴 → (𝑏 = 𝑎𝑏𝐴))
2423con3rr3 149 . . . . . . . . . . . . . 14 𝑏𝐴 → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2524adantl 480 . . . . . . . . . . . . 13 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2625imp 443 . . . . . . . . . . . 12 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 = 𝑎)
27 velsn 4140 . . . . . . . . . . . 12 (𝑏 ∈ {𝑎} ↔ 𝑏 = 𝑎)
2826, 27sylnibr 317 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 ∈ {𝑎})
2922, 28eldifd 3550 . . . . . . . . . 10 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
3029ex 448 . . . . . . . . 9 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3121, 30sylbi 205 . . . . . . . 8 (𝑏𝐵 → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3231impcom 444 . . . . . . 7 ((𝑎𝐴𝑏𝐵) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
33 neleq1 2887 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) ↔ 𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎)))
34 fveq2 6088 . . . . . . . . . 10 (𝑦 = 𝑏 → ((𝑉 VDeg 𝐸)‘𝑦) = ((𝑉 VDeg 𝐸)‘𝑏))
3534eqeq2d 2619 . . . . . . . . 9 (𝑦 = 𝑏 → (((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦) ↔ ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏)))
3633, 35imbi12d 332 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦)) ↔ (𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏))))
3736rspcv 3277 . . . . . . 7 (𝑏 ∈ (𝑉 ∖ {𝑎}) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦)) → (𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏))))
3832, 37syl 17 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑦)) → (𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏))))
39 nnel 2891 . . . . . . . . 9 𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) ↔ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎))
40 frisusgra 26285 . . . . . . . . . . . . . . 15 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
41 nbgraeledg 25725 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸 → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) ↔ {𝑏, 𝑎} ∈ ran 𝐸))
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝑉 FriendGrph 𝐸 → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) ↔ {𝑏, 𝑎} ∈ ran 𝐸))
43 prcom 4210 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
4443eleq1i 2678 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸)
4542, 44syl6bb 274 . . . . . . . . . . . . 13 (𝑉 FriendGrph 𝐸 → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) ↔ {𝑎, 𝑏} ∈ ran 𝐸))
4645biimpa 499 . . . . . . . . . . . 12 ((𝑉 FriendGrph 𝐸𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎)) → {𝑎, 𝑏} ∈ ran 𝐸)
4746a1d 25 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎)) → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))
4847expcom 449 . . . . . . . . . 10 (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸)))
4948a1d 25 . . . . . . . . 9 (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
5039, 49sylbi 205 . . . . . . . 8 𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
51 eqneqall 2792 . . . . . . . . . 10 (((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏) → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))
5251a1d 25 . . . . . . . . 9 (((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸)))
5352a1d 25 . . . . . . . 8 (((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏) → ((𝑎𝐴𝑏𝐵) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
5450, 53ja 171 . . . . . . 7 ((𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏)) → ((𝑎𝐴𝑏𝐵) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
5554com12 32 . . . . . 6 ((𝑎𝐴𝑏𝐵) → ((𝑏 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑎) → ((𝑉 VDeg 𝐸)‘𝑎) = ((𝑉 VDeg 𝐸)‘𝑏)) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
5618, 38, 553syld 57 . . . . 5 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) → (𝑉 FriendGrph 𝐸 → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
5756com3l 86 . . . 4 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)) → (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑏𝐵) → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸))))
584, 57mpcom 37 . . 3 (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑏𝐵) → (((𝑉 VDeg 𝐸)‘𝑎) ≠ ((𝑉 VDeg 𝐸)‘𝑏) → {𝑎, 𝑏} ∈ ran 𝐸)))
593, 58mpdi 43 . 2 (𝑉 FriendGrph 𝐸 → ((𝑎𝐴𝑏𝐵) → {𝑎, 𝑏} ∈ ran 𝐸))
6059ralrimivv 2952 1 (𝑉 FriendGrph 𝐸 → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  {crab 2899  cdif 3536  {csn 4124  {cpr 4126  cop 4130   class class class wbr 4577  ran crn 5029  cfv 5790  (class class class)co 6527   USGrph cusg 25625   Neighbors cnbgra 25712   VDeg cvdg 26186   FriendGrph cfrgra 26281
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 2032  ax-13 2232  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-xadd 11779  df-fz 12153  df-hash 12935  df-usgra 25628  df-nbgra 25715  df-vdgr 26187  df-frgra 26282
This theorem is referenced by:  frgrawopreglem5  26341  frgrawopreg1  26343  frgrawopreg2  26344
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