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Theorem tpsuni 22789
Description: The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Baseβ€˜πΎ)
istps.j 𝐽 = (TopOpenβ€˜πΎ)
Assertion
Ref Expression
tpsuni (𝐾 ∈ TopSp β†’ 𝐴 = βˆͺ 𝐽)

Proof of Theorem tpsuni
StepHypRef Expression
1 istps.a . . 3 𝐴 = (Baseβ€˜πΎ)
2 istps.j . . 3 𝐽 = (TopOpenβ€˜πΎ)
31, 2istps2 22788 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = βˆͺ 𝐽))
43simprbi 496 1 (𝐾 ∈ TopSp β†’ 𝐴 = βˆͺ 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4902  β€˜cfv 6536  Basecbs 17151  TopOpenctopn 17374  Topctop 22746  TopSpctps 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-top 22747  df-topon 22764  df-topsp 22786
This theorem is referenced by:  mreclatdemoBAD  22951  haustsms  23991  cnextucn  24159  ressxms  24385  rlmbn  25240  rrhf  33508  esumcocn  33608  sibf0  33863  sibfof  33869  sitgclg  33871  sitgaddlemb  33877  sitmcl  33880  binomcxplemdvbinom  43669  binomcxplemnotnn0  43672  qndenserrn  45568
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