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Theorem istopg 12176
 Description: Express the predicate " is a topology". See istopfin 12177 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
istopg
Distinct variable groups:   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem istopg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pweq 3513 . . . . 5
2 eleq2 2203 . . . . 5
31, 2raleqbidv 2638 . . . 4
4 eleq2 2203 . . . . . 6
54raleqbi1dv 2634 . . . . 5
65raleqbi1dv 2634 . . . 4
73, 6anbi12d 464 . . 3
8 df-top 12175 . . 3
97, 8elab2g 2831 . 2
10 df-ral 2421 . . . 4
11 elpw2g 4081 . . . . . 6
1211imbi1d 230 . . . . 5
1312albidv 1796 . . . 4
1410, 13syl5bb 191 . . 3
1514anbi1d 460 . 2
169, 15bitrd 187 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1329   wceq 1331   wcel 1480  wral 2416   cin 3070   wss 3071  cpw 3510  cuni 3736  ctop 12174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-top 12175 This theorem is referenced by:  istopfin  12177  uniopn  12178  inopn  12180  tgcl  12243  distop  12264  epttop  12269
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