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Theorem topnpropd 13664
Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
Hypotheses
Ref Expression
topnpropd.1  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
topnpropd.2  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L )
)
Assertion
Ref Expression
topnpropd  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )

Proof of Theorem topnpropd
StepHypRef Expression
1 topnpropd.2 . . 3  |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L )
)
2 topnpropd.1 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
31, 2oveq12d 6099 . 2  |-  ( ph  ->  ( (TopSet `  K
)t  ( Base `  K
) )  =  ( (TopSet `  L )t  ( Base `  L ) ) )
4 eqid 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2436 . . 3  |-  (TopSet `  K )  =  (TopSet `  K )
64, 5topnval 13662 . 2  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
7 eqid 2436 . . 3  |-  ( Base `  L )  =  (
Base `  L )
8 eqid 2436 . . 3  |-  (TopSet `  L )  =  (TopSet `  L )
97, 8topnval 13662 . 2  |-  ( (TopSet `  L )t  ( Base `  L
) )  =  (
TopOpen `  L )
103, 6, 93eqtr3g 2491 1  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   ` cfv 5454  (class class class)co 6081   Basecbs 13469  TopSetcts 13535   ↾t crest 13648   TopOpenctopn 13649
This theorem is referenced by:  sratopn  16256  tpsprop2d  17006  nrgtrg  18725  zhmnrg  24351
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-rest 13650  df-topn 13651
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