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Theorem topnpropd 14371
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 6108 . 2  |-  ( ph  ->  ( (TopSet `  K
)t  ( Base `  K
) )  =  ( (TopSet `  L )t  ( Base `  L ) ) )
4 eqid 2441 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2441 . . 3  |-  (TopSet `  K )  =  (TopSet `  K )
64, 5topnval 14369 . 2  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
7 eqid 2441 . . 3  |-  ( Base `  L )  =  (
Base `  L )
8 eqid 2441 . . 3  |-  (TopSet `  L )  =  (TopSet `  L )
97, 8topnval 14369 . 2  |-  ( (TopSet `  L )t  ( Base `  L
) )  =  (
TopOpen `  L )
103, 6, 93eqtr3g 2496 1  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   ` cfv 5415  (class class class)co 6090   Basecbs 14170  TopSetcts 14240   ↾t crest 14355   TopOpenctopn 14356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-rest 14357  df-topn 14358
This theorem is referenced by:  sratopn  17244  tpsprop2d  18505  nrgtrg  20229  zhmnrg  26332
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