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Theorem frgrancvvdeqlemC 25846
 Description: Lemma C for frgrancvvdeq 25849. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-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
frgrancvvdeqlemC
Distinct variable groups:   ,,   ,,   ,,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   (,)   (,)   ()

Proof of Theorem frgrancvvdeqlemC
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.nx . . 3 Neighbors
2 frgrancvvdeq.ny . . 3 Neighbors
3 frgrancvvdeq.x . . 3
4 frgrancvvdeq.y . . 3
5 frgrancvvdeq.ne . . 3
6 frgrancvvdeq.xy . . 3
7 frgrancvvdeq.f . . 3 FriendGrph
8 frgrancvvdeq.a . . 3
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem5 25841 . 2
107adantr 472 . . . . . . 7 FriendGrph
112eleq2i 2541 . . . . . . . . . 10 Neighbors
12 frisusgra 25799 . . . . . . . . . . 11 FriendGrph USGrph
13 nbgraisvtx 25238 . . . . . . . . . . 11 USGrph Neighbors
147, 12, 133syl 18 . . . . . . . . . 10 Neighbors
1511, 14syl5bi 225 . . . . . . . . 9
1615imp 436 . . . . . . . 8
173adantr 472 . . . . . . . 8
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 25838 . . . . . . . . . 10
19 df-nel 2644 . . . . . . . . . . 11
20 eleq1 2537 . . . . . . . . . . . . . 14
2120biimpcd 232 . . . . . . . . . . . . 13
2221con3rr3 143 . . . . . . . . . . . 12
23 df-ne 2643 . . . . . . . . . . . 12
2422, 23syl6ibr 235 . . . . . . . . . . 11
2519, 24sylbi 200 . . . . . . . . . 10
2618, 25syl 17 . . . . . . . . 9
2726imp 436 . . . . . . . 8
2816, 17, 273jca 1210 . . . . . . 7
2910, 28jca 541 . . . . . 6 FriendGrph
30 frgraun 25803 . . . . . . 7 FriendGrph
3130imp 436 . . . . . 6 FriendGrph
32 reurex 2995 . . . . . . 7
33 df-rex 2762 . . . . . . 7
3432, 33sylib 201 . . . . . 6
3529, 31, 343syl 18 . . . . 5
367, 12syl 17 . . . . . . . 8 USGrph
37 simprrr 783 . . . . . . . . . . . 12 USGrph
381eleq2i 2541 . . . . . . . . . . . . 13 Neighbors
39 nbgraeledg 25237 . . . . . . . . . . . . . . 15 USGrph Neighbors
4039ad2antlr 741 . . . . . . . . . . . . . 14 USGrph Neighbors
4140adantr 472 . . . . . . . . . . . . 13 USGrph Neighbors
4238, 41syl5bb 265 . . . . . . . . . . . 12 USGrph
4337, 42mpbird 240 . . . . . . . . . . 11 USGrph
44 nbgraeledg 25237 . . . . . . . . . . . . . . . . . 18 USGrph Neighbors
4544biimprcd 233 . . . . . . . . . . . . . . . . 17 USGrph Neighbors
4645adantr 472 . . . . . . . . . . . . . . . 16 USGrph Neighbors
4746adantl 473 . . . . . . . . . . . . . . 15 USGrph Neighbors
4847com12 31 . . . . . . . . . . . . . 14 USGrph Neighbors
4948ad2antlr 741 . . . . . . . . . . . . 13 USGrph Neighbors
5049imp 436 . . . . . . . . . . . 12 USGrph Neighbors
51 elin 3608 . . . . . . . . . . . . . . . 16 Neighbors Neighbors
52 simpll 768 . . . . . . . . . . . . . . . . . . . . 21 USGrph
5339bicomd 206 . . . . . . . . . . . . . . . . . . . . . . . 24 USGrph Neighbors
5453adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 USGrph Neighbors
5554biimpa 492 . . . . . . . . . . . . . . . . . . . . . 22 USGrph Neighbors
5655, 38sylibr 217 . . . . . . . . . . . . . . . . . . . . 21 USGrph
5752, 56jca 541 . . . . . . . . . . . . . . . . . . . 20 USGrph
58 preq1 4042 . . . . . . . . . . . . . . . . . . . . . . . . 25
5958eleq1d 2533 . . . . . . . . . . . . . . . . . . . . . . . 24
6059riotabidv 6272 . . . . . . . . . . . . . . . . . . . . . . 23
6160cbvmptv 4488 . . . . . . . . . . . . . . . . . . . . . 22
628, 61eqtri 2493 . . . . . . . . . . . . . . . . . . . . 21
631, 2, 3, 4, 5, 6, 7, 62frgrancvvdeqlem6 25842 . . . . . . . . . . . . . . . . . . . 20 Neighbors
64 eleq2 2538 . . . . . . . . . . . . . . . . . . . . . 22 Neighbors Neighbors
6564eqcoms 2479 . . . . . . . . . . . . . . . . . . . . 21 Neighbors Neighbors
66 elsni 3985 . . . . . . . . . . . . . . . . . . . . 21
6765, 66syl6bi 236 . . . . . . . . . . . . . . . . . . . 20 Neighbors Neighbors
6857, 63, 673syl 18 . . . . . . . . . . . . . . . . . . 19 USGrph Neighbors
6968expcom 442 . . . . . . . . . . . . . . . . . 18 USGrph Neighbors
7069ad2antll 743 . . . . . . . . . . . . . . . . 17 USGrph Neighbors
7170com3r 81 . . . . . . . . . . . . . . . 16 Neighbors USGrph
7251, 71sylbir 218 . . . . . . . . . . . . . . 15 Neighbors USGrph
7372ex 441 . . . . . . . . . . . . . 14 Neighbors USGrph
7473com14 90 . . . . . . . . . . . . 13 USGrph Neighbors
7574imp31 439 . . . . . . . . . . . 12 USGrph Neighbors
7650, 75mpd 15 . . . . . . . . . . 11 USGrph
7743, 76jca 541 . . . . . . . . . 10 USGrph
7877ex 441 . . . . . . . . 9 USGrph
7978ex 441 . . . . . . . 8 USGrph
8036, 79mpdan 681 . . . . . . 7
8180imp 436 . . . . . 6
8281eximdv 1772 . . . . 5
8335, 82mpd 15 . . . 4
84 df-rex 2762 . . . 4
8583, 84sylibr 217 . . 3
8685ralrimiva 2809 . 2
87 dffo3 6052 . 2
889, 86, 87sylanbrc 677 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   w3a 1007   wceq 1452  wex 1671   wcel 1904   wne 2641   wnel 2642  wral 2756  wrex 2757  wreu 2758   cin 3389  csn 3959  cpr 3961  cop 3965   class class class wbr 4395   cmpt 4454   crn 4840  wf 5585  wfo 5587  cfv 5589  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-rmo 2764  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:  frgrancvvdeqlem8  25847
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