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Theorem fneval 26381
 Description: Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fneval.1
Assertion
Ref Expression
fneval

Proof of Theorem fneval
StepHypRef Expression
1 fneval.1 . . . 4
21breqi 4221 . . 3
3 brin 4262 . . . 4
4 fnerel 26361 . . . . . 6
54relbrcnv 5248 . . . . 5
65anbi2i 677 . . . 4
73, 6bitri 242 . . 3
82, 7bitri 242 . 2
9 eqid 2438 . . . . . 6
10 eqid 2438 . . . . . 6
119, 10isfne4b 26364 . . . . 5
1210, 9isfne4b 26364 . . . . . 6
13 eqcom 2440 . . . . . . 7
1413anbi1i 678 . . . . . 6
1512, 14syl6bb 254 . . . . 5
1611, 15bi2anan9r 846 . . . 4
17 eqss 3365 . . . . . 6
1817anbi2i 677 . . . . 5
19 anandi 803 . . . . 5
2018, 19bitri 242 . . . 4
2116, 20syl6bbr 256 . . 3
22 unieq 4026 . . . . 5
23 unitg 17037 . . . . . 6
24 unitg 17037 . . . . . 6
2523, 24eqeqan12d 2453 . . . . 5
2622, 25syl5ib 212 . . . 4
2726pm4.71rd 618 . . 3
2821, 27bitr4d 249 . 2
298, 28syl5bb 250 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726   cin 3321   wss 3322  cuni 4017   class class class wbr 4215  ccnv 4880  cfv 5457  ctg 13670  cfne 26353 This theorem is referenced by:  fneer  26382  topfneec  26385  topfneec2  26386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-topgen 13672  df-fne 26357
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