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Theorem frgrancvvdeqlem9 26568
 Description: Lemma 9 for frgrancvvdeq 26569. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
Assertion
Ref Expression
frgrancvvdeqlem9 (𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
Distinct variable groups:   𝑓,𝐸,𝑥,𝑦   𝑓,𝑉,𝑥,𝑦

Proof of Theorem frgrancvvdeqlem9
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . . . 6 (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∈ V
2 mptexg 6389 . . . . . 6 ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∈ V → (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)) ∈ V)
31, 2mp1i 13 . . . . 5 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)) ∈ V)
4 eqid 2610 . . . . . 6 (⟨𝑉, 𝐸⟩ Neighbors 𝑥) = (⟨𝑉, 𝐸⟩ Neighbors 𝑥)
5 eqid 2610 . . . . . 6 (⟨𝑉, 𝐸⟩ Neighbors 𝑦) = (⟨𝑉, 𝐸⟩ Neighbors 𝑦)
6 simpllr 795 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → 𝑥𝑉)
7 eldifi 3694 . . . . . . 7 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
87ad2antlr 759 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → 𝑦𝑉)
9 eldif 3550 . . . . . . . 8 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
10 velsn 4141 . . . . . . . . . . . 12 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1110biimpri 217 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1211equcoms 1934 . . . . . . . . . 10 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1312necon3bi 2808 . . . . . . . . 9 𝑦 ∈ {𝑥} → 𝑥𝑦)
1413adantl 481 . . . . . . . 8 ((𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}) → 𝑥𝑦)
159, 14sylbi 206 . . . . . . 7 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1615ad2antlr 759 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → 𝑥𝑦)
17 simpr 476 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥))
18 simplll 794 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → 𝑉 FriendGrph 𝐸)
19 eqid 2610 . . . . . 6 (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)) = (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸))
204, 5, 6, 8, 16, 17, 18, 19frgrancvvdeqlem8 26567 . . . . 5 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)):(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦))
21 f1oeq1 6040 . . . . . 6 (𝑓 = (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)) → (𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦) ↔ (𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)):(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
2221spcegv 3267 . . . . 5 ((𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)) ∈ V → ((𝑢 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) ↦ (𝑤 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑦){𝑢, 𝑤} ∈ ran 𝐸)):(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
233, 20, 22sylc 63 . . . 4 ((((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) ∧ 𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥)) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦))
2423ex 449 . . 3 (((𝑉 FriendGrph 𝐸𝑥𝑉) ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → (𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
2524ralrimiva 2949 . 2 ((𝑉 FriendGrph 𝐸𝑥𝑉) → ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
2625ralrimiva 2949 1 (𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  –1-1-onto→wf1o 5803  ℩crio 6510  (class class class)co 6549   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-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-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:  frgrancvvdeq  26569  frgrancvvdgeq  26570
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