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Theorem tpsuni 21544
Description: The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Base‘𝐾)
istps.j 𝐽 = (TopOpen‘𝐾)
Assertion
Ref Expression
tpsuni (𝐾 ∈ TopSp → 𝐴 = 𝐽)

Proof of Theorem tpsuni
StepHypRef Expression
1 istps.a . . 3 𝐴 = (Base‘𝐾)
2 istps.j . . 3 𝐽 = (TopOpen‘𝐾)
31, 2istps2 21543 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
43simprbi 500 1 (𝐾 ∈ TopSp → 𝐴 = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112   cuni 4803  cfv 6328  Basecbs 16478  TopOpenctopn 16690  Topctop 21501  TopSpctps 21540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-top 21502  df-topon 21519  df-topsp 21541
This theorem is referenced by:  mreclatdemoBAD  21704  haustsms  22744  cnextucn  22912  ressxms  23135  rlmbn  23968  rrhf  31347  esumcocn  31447  sibf0  31700  sibfof  31706  sitgclg  31708  sitgaddlemb  31714  sitmcl  31717  binomcxplemdvbinom  41044  binomcxplemnotnn0  41047  qndenserrn  42928
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