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Theorem topnpropd 14709
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 6313 . 2  |-  ( ph  ->  ( (TopSet `  K
)t  ( Base `  K
) )  =  ( (TopSet `  L )t  ( Base `  L ) ) )
4 eqid 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2467 . . 3  |-  (TopSet `  K )  =  (TopSet `  K )
64, 5topnval 14707 . 2  |-  ( (TopSet `  K )t  ( Base `  K
) )  =  (
TopOpen `  K )
7 eqid 2467 . . 3  |-  ( Base `  L )  =  (
Base `  L )
8 eqid 2467 . . 3  |-  (TopSet `  L )  =  (TopSet `  L )
97, 8topnval 14707 . 2  |-  ( (TopSet `  L )t  ( Base `  L
) )  =  (
TopOpen `  L )
103, 6, 93eqtr3g 2531 1  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   ` cfv 5594  (class class class)co 6295   Basecbs 14507  TopSetcts 14578   ↾t crest 14693   TopOpenctopn 14694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-rest 14695  df-topn 14696
This theorem is referenced by:  sratopn  17702  tpsprop2d  19311  nrgtrg  21066  zhmnrg  27773
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