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Theorem istpsi 19207
Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istpsi.b  |-  ( Base `  K )  =  A
istpsi.j  |-  ( TopOpen `  K )  =  J
istpsi.1  |-  A  = 
U. J
istpsi.2  |-  J  e. 
Top
Assertion
Ref Expression
istpsi  |-  K  e. 
TopSp

Proof of Theorem istpsi
StepHypRef Expression
1 istpsi.2 . 2  |-  J  e. 
Top
2 istpsi.1 . 2  |-  A  = 
U. J
3 istpsi.b . . . 4  |-  ( Base `  K )  =  A
43eqcomi 2475 . . 3  |-  A  =  ( Base `  K
)
5 istpsi.j . . . 4  |-  ( TopOpen `  K )  =  J
65eqcomi 2475 . . 3  |-  J  =  ( TopOpen `  K )
74, 6istps2 19200 . 2  |-  ( K  e.  TopSp 
<->  ( J  e.  Top  /\  A  =  U. J
) )
81, 2, 7mpbir2an 913 1  |-  K  e. 
TopSp
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   U.cuni 4240   ` cfv 5581   Basecbs 14481   TopOpenctopn 14668   Topctop 19156   TopSpctps 19159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-top 19161  df-topon 19164  df-topsp 19165
This theorem is referenced by:  indistps2  19274
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