MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpspropd Structured version   Unicode version

Theorem tpspropd 18678
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 5804 . . 3  |-  ( ph  ->  (TopOn `  ( Base `  K ) )  =  (TopOn `  ( Base `  L ) ) )
41, 3eleq12d 2536 . 2  |-  ( ph  ->  ( ( TopOpen `  K
)  e.  (TopOn `  ( Base `  K )
)  <->  ( TopOpen `  L
)  e.  (TopOn `  ( Base `  L )
) ) )
5 eqid 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
6 eqid 2454 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
75, 6istps 18674 . 2  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
8 eqid 2454 . . 3  |-  ( Base `  L )  =  (
Base `  L )
9 eqid 2454 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
108, 9istps 18674 . 2  |-  ( L  e.  TopSp 
<->  ( TopOpen `  L )  e.  (TopOn `  ( Base `  L ) ) )
114, 7, 103bitr4g 288 1  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   ` cfv 5527   Basecbs 14293   TopOpenctopn 14480  TopOnctopon 18632   TopSpctps 18634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-top 18636  df-topon 18639  df-topsp 18640
This theorem is referenced by:  tpsprop2d  18679  xmspropd  20181
  Copyright terms: Public domain W3C validator