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Theorem frgrancvvdeqlem3 26559
Description: Lemma 3 for frgrancvvdeq 26569. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgrancvvdeq.ny 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
frgrancvvdeq.x (𝜑𝑋𝑉)
frgrancvvdeq.y (𝜑𝑌𝑉)
frgrancvvdeq.ne (𝜑𝑋𝑌)
frgrancvvdeq.xy (𝜑𝑌𝐷)
frgrancvvdeq.f (𝜑𝑉 FriendGrph 𝐸)
frgrancvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
Assertion
Ref Expression
frgrancvvdeqlem3 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)
Distinct variable groups:   𝑦,𝐷   𝑥,𝑦,𝑉   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐷(𝑥)   𝑁(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlem3
StepHypRef Expression
1 frgrancvvdeq.f . . . 4 (𝜑𝑉 FriendGrph 𝐸)
21adantr 480 . . 3 ((𝜑𝑥𝐷) → 𝑉 FriendGrph 𝐸)
3 frgrancvvdeq.nx . . . . . . 7 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
43eleq2i 2680 . . . . . 6 (𝑥𝐷𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
5 frisusgra 26519 . . . . . . 7 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
6 nbgraisvtx 25960 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑥𝑉))
71, 5, 63syl 18 . . . . . 6 (𝜑 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑥𝑉))
84, 7syl5bi 231 . . . . 5 (𝜑 → (𝑥𝐷𝑥𝑉))
98imp 444 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑉)
10 frgrancvvdeq.y . . . . 5 (𝜑𝑌𝑉)
1110adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑌𝑉)
12 frgrancvvdeq.ny . . . . . 6 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
13 frgrancvvdeq.x . . . . . 6 (𝜑𝑋𝑉)
14 frgrancvvdeq.ne . . . . . 6 (𝜑𝑋𝑌)
15 frgrancvvdeq.xy . . . . . 6 (𝜑𝑌𝐷)
16 frgrancvvdeq.a . . . . . 6 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
173, 12, 13, 10, 14, 15, 1, 16frgrancvvdeqlem1 26557 . . . . 5 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
18 eldif 3550 . . . . . 6 (𝑌 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑌𝑉 ∧ ¬ 𝑌 ∈ {𝑥}))
19 vsnid 4156 . . . . . . . . 9 𝑥 ∈ {𝑥}
20 eleq1 2676 . . . . . . . . . 10 (𝑌 = 𝑥 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2120eqcoms 2618 . . . . . . . . 9 (𝑥 = 𝑌 → (𝑌 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
2219, 21mpbiri 247 . . . . . . . 8 (𝑥 = 𝑌𝑌 ∈ {𝑥})
2322necon3bi 2808 . . . . . . 7 𝑌 ∈ {𝑥} → 𝑥𝑌)
2423adantl 481 . . . . . 6 ((𝑌𝑉 ∧ ¬ 𝑌 ∈ {𝑥}) → 𝑥𝑌)
2518, 24sylbi 206 . . . . 5 (𝑌 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑌)
2617, 25syl 17 . . . 4 ((𝜑𝑥𝐷) → 𝑥𝑌)
279, 11, 263jca 1235 . . 3 ((𝜑𝑥𝐷) → (𝑥𝑉𝑌𝑉𝑥𝑌))
28 frgraunss 26522 . . 3 (𝑉 FriendGrph 𝐸 → ((𝑥𝑉𝑌𝑉𝑥𝑌) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸))
292, 27, 28sylc 63 . 2 ((𝜑𝑥𝐷) → ∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)
30 prex 4836 . . . . . . . . . . . 12 {𝑥, 𝑦} ∈ V
31 prex 4836 . . . . . . . . . . . 12 {𝑦, 𝑌} ∈ V
3230, 31prss 4291 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑌} ∈ ran 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)
33 simpr 476 . . . . . . . . . . 11 (({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑌} ∈ ran 𝐸) → {𝑦, 𝑌} ∈ ran 𝐸)
3432, 33sylbir 224 . . . . . . . . . 10 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸 → {𝑦, 𝑌} ∈ ran 𝐸)
3534ad2antll 761 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)) → {𝑦, 𝑌} ∈ ran 𝐸)
3612a1i 11 . . . . . . . . . . . . 13 (𝜑𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
3736eleq2d 2673 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌)))
38 nbgraeledg 25959 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) ↔ {𝑦, 𝑌} ∈ ran 𝐸))
391, 5, 383syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) ↔ {𝑦, 𝑌} ∈ ran 𝐸))
4037, 39bitrd 267 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ ran 𝐸))
4140adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ ran 𝐸))
4241adantr 480 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)) → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ ran 𝐸))
4335, 42mpbird 246 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)) → 𝑦𝑁)
44 simpl 472 . . . . . . . . . 10 (({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑌} ∈ ran 𝐸) → {𝑥, 𝑦} ∈ ran 𝐸)
4532, 44sylbir 224 . . . . . . . . 9 ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸 → {𝑥, 𝑦} ∈ ran 𝐸)
4645ad2antll 761 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)) → {𝑥, 𝑦} ∈ ran 𝐸)
4743, 46jca 553 . . . . . . 7 (((𝜑𝑥𝐷) ∧ (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
4847ex 449 . . . . . 6 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸) → (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
4912eleq2i 2680 . . . . . . . . . . . . 13 (𝑦𝑁𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
5049, 39syl5bb 271 . . . . . . . . . . . 12 (𝜑 → (𝑦𝑁 ↔ {𝑦, 𝑌} ∈ ran 𝐸))
5150biimpd 218 . . . . . . . . . . 11 (𝜑 → (𝑦𝑁 → {𝑦, 𝑌} ∈ ran 𝐸))
5251adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑦𝑁 → {𝑦, 𝑌} ∈ ran 𝐸))
5352impcom 445 . . . . . . . . 9 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → {𝑦, 𝑌} ∈ ran 𝐸)
54 nbgraisvtx 25960 . . . . . . . . . . . . . . . 16 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → 𝑦𝑉))
551, 5, 543syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → 𝑦𝑉))
5649, 55syl5bi 231 . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑁𝑦𝑉))
5756adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → (𝑦𝑁𝑦𝑉))
5857impcom 445 . . . . . . . . . . . 12 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → 𝑦𝑉)
5958ad2antlr 759 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ ran 𝐸) → 𝑦𝑉)
60 simpl 472 . . . . . . . . . . . . 13 (({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → {𝑦, 𝑌} ∈ ran 𝐸)
61 id 22 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ∈ ran 𝐸 → {𝑥, 𝑦} ∈ ran 𝐸)
6260, 61anim12ci 589 . . . . . . . . . . . 12 ((({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ({𝑥, 𝑦} ∈ ran 𝐸 ∧ {𝑦, 𝑌} ∈ ran 𝐸))
6362, 32sylib 207 . . . . . . . . . . 11 ((({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ ran 𝐸) → {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)
6459, 63jca 553 . . . . . . . . . 10 ((({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) ∧ {𝑥, 𝑦} ∈ ran 𝐸) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸))
6564ex 449 . . . . . . . . 9 (({𝑦, 𝑌} ∈ ran 𝐸 ∧ (𝑦𝑁 ∧ (𝜑𝑥𝐷))) → ({𝑥, 𝑦} ∈ ran 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)))
6653, 65mpancom 700 . . . . . . . 8 ((𝑦𝑁 ∧ (𝜑𝑥𝐷)) → ({𝑥, 𝑦} ∈ ran 𝐸 → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)))
6766impancom 455 . . . . . . 7 ((𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → ((𝜑𝑥𝐷) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)))
6867com12 32 . . . . . 6 ((𝜑𝑥𝐷) → ((𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸) → (𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸)))
6948, 68impbid 201 . . . . 5 ((𝜑𝑥𝐷) → ((𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸) ↔ (𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
7069eubidv 2478 . . . 4 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸) ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
7170biimpd 218 . . 3 ((𝜑𝑥𝐷) → (∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸) → ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))
72 df-reu 2903 . . 3 (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸 ↔ ∃!𝑦(𝑦𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸))
73 df-reu 2903 . . 3 (∃!𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸 ↔ ∃!𝑦(𝑦𝑁 ∧ {𝑥, 𝑦} ∈ ran 𝐸))
7471, 72, 733imtr4g 284 . 2 ((𝜑𝑥𝐷) → (∃!𝑦𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ ran 𝐸 → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
7529, 74mpd 15 1 ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  ∃!weu 2458  wne 2780  wnel 2781  ∃!wreu 2898  cdif 3537  wss 3540  {csn 4125  {cpr 4127  cop 4131   class class class wbr 4583  cmpt 4643  ran crn 5039  crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946   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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-nbgra 25949  df-frgra 26516
This theorem is referenced by:  frgrancvvdeqlem4  26560  frgrancvvdeqlem5  26561
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