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Theorem frgrancvvdeqlem7 26563
 Description: Lemma 7 for frgrancvvdeq 26569. (Contributed by Alexander van der Vekens, 23-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
frgrancvvdeqlem7 ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸)
Distinct variable groups:   𝑦,𝐷,𝑥   𝑥,𝑉,𝑦   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlem7
StepHypRef Expression
1 frgrancvvdeq.nx . . 3 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
2 frgrancvvdeq.ny . . 3 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
3 frgrancvvdeq.x . . 3 (𝜑𝑋𝑉)
4 frgrancvvdeq.y . . 3 (𝜑𝑌𝑉)
5 frgrancvvdeq.ne . . 3 (𝜑𝑋𝑌)
6 frgrancvvdeq.xy . . 3 (𝜑𝑌𝐷)
7 frgrancvvdeq.f . . 3 (𝜑𝑉 FriendGrph 𝐸)
8 frgrancvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem6 26562 . 2 ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁))
10 fvex 6113 . . . 4 (𝐴𝑥) ∈ V
1110snid 4155 . . 3 (𝐴𝑥) ∈ {(𝐴𝑥)}
12 eleq2 2677 . . . . 5 ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} ↔ (𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁)))
1312biimpd 218 . . . 4 ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝐴𝑥) ∈ {(𝐴𝑥)} → (𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁)))
14 elin 3758 . . . . 5 ((𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) ↔ ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∧ (𝐴𝑥) ∈ 𝑁))
15 frisusgra 26519 . . . . . . . . 9 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
16 nbgraeledg 25959 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↔ {(𝐴𝑥), 𝑥} ∈ ran 𝐸))
17 prcom 4211 . . . . . . . . . . . 12 {(𝐴𝑥), 𝑥} = {𝑥, (𝐴𝑥)}
1817eleq1i 2679 . . . . . . . . . . 11 ({(𝐴𝑥), 𝑥} ∈ ran 𝐸 ↔ {𝑥, (𝐴𝑥)} ∈ ran 𝐸)
1916, 18syl6bb 275 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↔ {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2019biimpd 218 . . . . . . . . 9 (𝑉 USGrph 𝐸 → ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
217, 15, 203syl 18 . . . . . . . 8 (𝜑 → ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2221adantr 480 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2322com12 32 . . . . . 6 ((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2423adantr 480 . . . . 5 (((𝐴𝑥) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∧ (𝐴𝑥) ∈ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2514, 24sylbi 206 . . . 4 ((𝐴𝑥) ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
2613, 25syl6com 36 . . 3 ((𝐴𝑥) ∈ {(𝐴𝑥)} → ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸)))
2711, 26ax-mp 5 . 2 ({(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁) → ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸))
289, 27mpcom 37 1 ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781   ∩ cin 3539  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  ℩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:  frgrancvvdeqlemB  26565
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