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Theorem lmbr 17276
 Description: Express the binary relation "sequence converges to point " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 17247. (Contributed by Mario Carneiro, 14-Nov-2013.)
Hypothesis
Ref Expression
lmbr.2 TopOn
Assertion
Ref Expression
lmbr
Distinct variable groups:   ,,   ,,   ,   ,   ,,
Allowed substitution hints:   ()   ()

Proof of Theorem lmbr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmbr.2 . . . 4 TopOn
2 lmfval 17250 . . . 4 TopOn
31, 2syl 16 . . 3
43breqd 4183 . 2
5 reseq1 5099 . . . . . . . . 9
65feq1d 5539 . . . . . . . 8
76rexbidv 2687 . . . . . . 7
87imbi2d 308 . . . . . 6
98ralbidv 2686 . . . . 5
10 eleq1 2464 . . . . . . 7
1110imbi1d 309 . . . . . 6
1211ralbidv 2686 . . . . 5
139, 12sylan9bb 681 . . . 4
14 df-3an 938 . . . . 5
1514opabbii 4232 . . . 4
1613, 15brab2ga 4910 . . 3
17 df-3an 938 . . 3
1816, 17bitr4i 244 . 2
194, 18syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1649   wcel 1721  wral 2666  wrex 2667   class class class wbr 4172  copab 4225   crn 4838   cres 4839  wf 5409  cfv 5413  (class class class)co 6040   cpm 6978  cc 8944  cuz 10444  TopOnctopon 16914  clm 17244 This theorem is referenced by:  lmbr2  17277  lmfpm  17313  lmcl  17315  lmff  17319  lmmbr  19164 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-top 16918  df-topon 16921  df-lm 17247
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