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Theorem tpspropd 16997
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 5724 . . 3  |-  ( ph  ->  (TopOn `  ( Base `  K ) )  =  (TopOn `  ( Base `  L ) ) )
41, 3eleq12d 2503 . 2  |-  ( ph  ->  ( ( TopOpen `  K
)  e.  (TopOn `  ( Base `  K )
)  <->  ( TopOpen `  L
)  e.  (TopOn `  ( Base `  L )
) ) )
5 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 eqid 2435 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
75, 6istps 16993 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
8 eqid 2435 . . 3  |-  ( Base `  L )  =  (
Base `  L )
9 eqid 2435 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
108, 9istps 16993 . 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 1652    e. wcel 1725   ` cfv 5446   Basecbs 13461   TopOpenctopn 13641  TopOnctopon 16951   TopSpctps 16953
This theorem is referenced by:  tpsprop2d  16998  xmspropd  18495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-topon 16958  df-topsp 16959
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