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Theorem tpspropd 16678
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 5529 . . 3  |-  ( ph  ->  (TopOn `  ( Base `  K ) )  =  (TopOn `  ( Base `  L ) ) )
41, 3eleq12d 2351 . 2  |-  ( ph  ->  ( ( TopOpen `  K
)  e.  (TopOn `  ( Base `  K )
)  <->  ( TopOpen `  L
)  e.  (TopOn `  ( Base `  L )
) ) )
5 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 eqid 2283 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
75, 6istps 16674 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
8 eqid 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
9 eqid 2283 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
108, 9istps 16674 . 2  |-  ( L  e.  TopSp 
<->  ( TopOpen `  L )  e.  (TopOn `  ( Base `  L ) ) )
114, 7, 103bitr4g 279 1  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   ` cfv 5255   Basecbs 13148   TopOpenctopn 13326  TopOnctopon 16632   TopSpctps 16634
This theorem is referenced by:  tpsprop2d  16679  xmspropd  18019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639  df-topsp 16640
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