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Theorem tpspropd 16930
Description: A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tpspropd.1  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
tpspropd.2  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Assertion
Ref Expression
tpspropd  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )

Proof of Theorem tpspropd
StepHypRef Expression
1 tpspropd.2 . . 3  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
2 tpspropd.1 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
32fveq2d 5674 . . 3  |-  ( ph  ->  (TopOn `  ( Base `  K ) )  =  (TopOn `  ( Base `  L ) ) )
41, 3eleq12d 2457 . 2  |-  ( ph  ->  ( ( TopOpen `  K
)  e.  (TopOn `  ( Base `  K )
)  <->  ( TopOpen `  L
)  e.  (TopOn `  ( Base `  L )
) ) )
5 eqid 2389 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 eqid 2389 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
75, 6istps 16926 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
8 eqid 2389 . . 3  |-  ( Base `  L )  =  (
Base `  L )
9 eqid 2389 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
108, 9istps 16926 . 2  |-  ( L  e.  TopSp 
<->  ( TopOpen `  L )  e.  (TopOn `  ( Base `  L ) ) )
114, 7, 103bitr4g 280 1  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   ` cfv 5396   Basecbs 13398   TopOpenctopn 13578  TopOnctopon 16884   TopSpctps 16886
This theorem is referenced by:  tpsprop2d  16931  xmspropd  18395
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-top 16888  df-topon 16891  df-topsp 16892
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