MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  toponunii Structured version   Visualization version   GIF version

Theorem toponunii 21525
Description: The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
topontopi.1 𝐽 ∈ (TopOn‘𝐵)
Assertion
Ref Expression
toponunii 𝐵 = 𝐽

Proof of Theorem toponunii
StepHypRef Expression
1 topontopi.1 . 2 𝐽 ∈ (TopOn‘𝐵)
2 toponuni 21523 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
31, 2ax-mp 5 1 𝐵 = 𝐽
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112   cuni 4803  cfv 6328  TopOnctopon 21519
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-topon 21520
This theorem is referenced by:  toponrestid  21530  indisuni  21612  indistpsx  21619  letopuni  21816  dfac14  22227  unicntop  23395  sszcld  23426  reperflem  23427  cnperf  23429  iiuni  23490  abscncfALT  23533  cncfcnvcn  23534  cnheiborlem  23563  cnheibor  23564  cnllycmp  23565  bndth  23567  mbfimaopnlem  24263  limcnlp  24485  limcflflem  24487  limcflf  24488  limcmo  24489  limcres  24493  limccnp  24498  limccnp2  24499  perfdvf  24510  recnperf  24512  dvcnp2  24527  dvaddbr  24545  dvmulbr  24546  dvcobr  24553  dvcnvlem  24583  lhop1lem  24620  taylthlem2  24973  abelth  25040  cxpcn3  25341  lgamucov  25627  ftalem3  25664  blocni  28592  ipasslem8  28624  ubthlem1  28657  tpr2uni  31262  tpr2rico  31269  mndpluscn  31283  rmulccn  31285  raddcn  31286  cvxsconn  32604  cvmlift2lem11  32674  ivthALT  33797  poimir  35089  broucube  35090  dvtanlem  35105  ftc1cnnc  35128  dvasin  35140  dvacos  35141  dvreasin  35142  dvreacos  35143  areacirclem2  35145  reheibor  35276  islptre  42254  dirkercncf  42742  fourierdlem62  42803
  Copyright terms: Public domain W3C validator