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Theorem frgrancvvdeqlem4 25840
 Description: Lemma 4 for frgrancvvdeq 25849. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx Neighbors
frgrancvvdeq.ny Neighbors
frgrancvvdeq.x
frgrancvvdeq.y
frgrancvvdeq.ne
frgrancvvdeq.xy
frgrancvvdeq.f FriendGrph
frgrancvvdeq.a
Assertion
Ref Expression
frgrancvvdeqlem4 Neighbors
Distinct variable groups:   ,   ,,   ,,   ,   ,   ,
Allowed substitution hints:   ()   (,)   ()   ()   (,)   ()

Proof of Theorem frgrancvvdeqlem4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.ny . . 3 Neighbors
21ineq2i 3622 . 2 Neighbors Neighbors Neighbors
3 frgrancvvdeq.nx . . . . . . 7 Neighbors
43eleq2i 2541 . . . . . 6 Neighbors
5 frgrancvvdeq.f . . . . . . . 8 FriendGrph
6 frisusgra 25799 . . . . . . . 8 FriendGrph USGrph
75, 6syl 17 . . . . . . 7 USGrph
8 nbgraisvtx 25238 . . . . . . 7 USGrph Neighbors
97, 8syl 17 . . . . . 6 Neighbors
104, 9syl5bi 225 . . . . 5
1110imp 436 . . . 4
12 frgrancvvdeq.x . . . . 5
13 frgrancvvdeq.y . . . . 5
14 frgrancvvdeq.ne . . . . 5
15 frgrancvvdeq.xy . . . . 5
16 frgrancvvdeq.a . . . . 5
173, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem1 25837 . . . 4
185adantr 472 . . . . 5 FriendGrph
19 frisusgranb 25804 . . . . 5 FriendGrph Neighbors Neighbors
2018, 19syl 17 . . . 4 Neighbors Neighbors
2111, 17, 20jca31 543 . . 3 Neighbors Neighbors
22 sneq 3969 . . . . . . . . 9
2322difeq2d 3540 . . . . . . . 8
24 oveq2 6316 . . . . . . . . . . 11 Neighbors Neighbors
2524ineq1d 3624 . . . . . . . . . 10 Neighbors Neighbors Neighbors Neighbors
2625eqeq1d 2473 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors
2726rexbidv 2892 . . . . . . . 8 Neighbors Neighbors Neighbors Neighbors
2823, 27raleqbidv 2987 . . . . . . 7 Neighbors Neighbors Neighbors Neighbors
2928rspcva 3134 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
30 oveq2 6316 . . . . . . . . . . . 12 Neighbors Neighbors
3130ineq2d 3625 . . . . . . . . . . 11 Neighbors Neighbors Neighbors Neighbors
3231eqeq1d 2473 . . . . . . . . . 10 Neighbors Neighbors Neighbors Neighbors
3332rexbidv 2892 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors
3433rspcva 3134 . . . . . . . 8 Neighbors Neighbors Neighbors Neighbors
35 ssnid 3989 . . . . . . . . . . . . . . 15
36 eleq2 2538 . . . . . . . . . . . . . . . . 17 Neighbors Neighbors Neighbors Neighbors
3736eqcoms 2479 . . . . . . . . . . . . . . . 16 Neighbors Neighbors Neighbors Neighbors
38 elin 3608 . . . . . . . . . . . . . . . . 17 Neighbors Neighbors Neighbors Neighbors
3938biimpi 199 . . . . . . . . . . . . . . . 16 Neighbors Neighbors Neighbors Neighbors
4037, 39syl6bi 236 . . . . . . . . . . . . . . 15 Neighbors Neighbors Neighbors Neighbors
4135, 40mpi 20 . . . . . . . . . . . . . 14 Neighbors Neighbors Neighbors Neighbors
42 nbgraeledg 25237 . . . . . . . . . . . . . . . . . . . . . 22 USGrph Neighbors
43 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . 23
4443eleq1i 2540 . . . . . . . . . . . . . . . . . . . . . 22
4542, 44syl6bb 269 . . . . . . . . . . . . . . . . . . . . 21 USGrph Neighbors
4645biimpd 212 . . . . . . . . . . . . . . . . . . . 20 USGrph Neighbors
477, 46syl 17 . . . . . . . . . . . . . . . . . . 19 Neighbors
4847adantr 472 . . . . . . . . . . . . . . . . . 18 Neighbors
4948com12 31 . . . . . . . . . . . . . . . . 17 Neighbors
5049adantr 472 . . . . . . . . . . . . . . . 16 Neighbors Neighbors
5150imp 436 . . . . . . . . . . . . . . 15 Neighbors Neighbors
521eqcomi 2480 . . . . . . . . . . . . . . . . . . 19 Neighbors
5352eleq2i 2541 . . . . . . . . . . . . . . . . . 18 Neighbors
5453biimpi 199 . . . . . . . . . . . . . . . . 17 Neighbors
5554adantl 473 . . . . . . . . . . . . . . . 16 Neighbors Neighbors
563, 1, 12, 13, 14, 15, 5, 16frgrancvvdeqlem3 25839 . . . . . . . . . . . . . . . 16
57 preq2 4043 . . . . . . . . . . . . . . . . . 18
5857eleq1d 2533 . . . . . . . . . . . . . . . . 17
5958riota2 6292 . . . . . . . . . . . . . . . 16
6055, 56, 59syl2an 485 . . . . . . . . . . . . . . 15 Neighbors Neighbors
6151, 60mpbid 215 . . . . . . . . . . . . . 14 Neighbors Neighbors
6241, 61sylan 479 . . . . . . . . . . . . 13 Neighbors Neighbors
6362eqcomd 2477 . . . . . . . . . . . 12 Neighbors Neighbors
6463sneqd 3971 . . . . . . . . . . 11 Neighbors Neighbors
65 eqeq1 2475 . . . . . . . . . . . 12 Neighbors Neighbors Neighbors Neighbors
6665adantr 472 . . . . . . . . . . 11 Neighbors Neighbors Neighbors Neighbors
6764, 66mpbird 240 . . . . . . . . . 10 Neighbors Neighbors Neighbors Neighbors
6867ex 441 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors
6968rexlimivw 2869 . . . . . . . 8 Neighbors Neighbors Neighbors Neighbors
7034, 69syl 17 . . . . . . 7 Neighbors Neighbors Neighbors Neighbors
7170expcom 442 . . . . . 6 Neighbors Neighbors Neighbors Neighbors
7229, 71syl 17 . . . . 5 Neighbors Neighbors Neighbors Neighbors
7372impancom 447 . . . 4 Neighbors Neighbors Neighbors Neighbors
7473imp 436 . . 3 Neighbors Neighbors Neighbors Neighbors
7521, 74mpcom 36 . 2 Neighbors Neighbors
762, 75syl5req 2518 1 Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904   wne 2641   wnel 2642  wral 2756  wrex 2757  wreu 2758   cdif 3387   cin 3389  csn 3959  cpr 3961  cop 3965   class class class wbr 4395   cmpt 4454   crn 4840  crio 6269  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224   FriendGrph cfrgra 25795 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-frgra 25796 This theorem is referenced by:  frgrancvvdeqlem6  25842
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