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Theorem tpsprop2d 22304
Description: A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tpsprop2d.1 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐ฟ))
tpsprop2d.2 (๐œ‘ โ†’ (TopSetโ€˜๐พ) = (TopSetโ€˜๐ฟ))
Assertion
Ref Expression
tpsprop2d (๐œ‘ โ†’ (๐พ โˆˆ TopSp โ†” ๐ฟ โˆˆ TopSp))

Proof of Theorem tpsprop2d
StepHypRef Expression
1 tpsprop2d.1 . 2 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐ฟ))
2 tpsprop2d.2 . . 3 (๐œ‘ โ†’ (TopSetโ€˜๐พ) = (TopSetโ€˜๐ฟ))
31, 2topnpropd 17323 . 2 (๐œ‘ โ†’ (TopOpenโ€˜๐พ) = (TopOpenโ€˜๐ฟ))
41, 3tpspropd 22303 1 (๐œ‘ โ†’ (๐พ โˆˆ TopSp โ†” ๐ฟ โˆˆ TopSp))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โ†” wb 205   = wceq 1542   โˆˆ wcel 2107  โ€˜cfv 6497  Basecbs 17088  TopSetcts 17144  TopSpctps 22297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-rest 17309  df-topn 17310  df-top 22259  df-topon 22276  df-topsp 22298
This theorem is referenced by: (None)
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