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

Theorem istopon 22414
Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
istopon (𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))

Proof of Theorem istopon
Dummy variables 𝑏 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6930 . 2 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 ∈ V)
2 uniexg 7730 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
3 eleq1 2822 . . . 4 (𝐡 = βˆͺ 𝐽 β†’ (𝐡 ∈ V ↔ βˆͺ 𝐽 ∈ V))
42, 3syl5ibrcom 246 . . 3 (𝐽 ∈ Top β†’ (𝐡 = βˆͺ 𝐽 β†’ 𝐡 ∈ V))
54imp 408 . 2 ((𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽) β†’ 𝐡 ∈ V)
6 eqeq1 2737 . . . . . 6 (𝑏 = 𝐡 β†’ (𝑏 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝑗))
76rabbidv 3441 . . . . 5 (𝑏 = 𝐡 β†’ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
8 df-topon 22413 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗})
9 vpwex 5376 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 5379 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 4070 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏 ↔ βˆ€π‘— ∈ Top (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4950 . . . . . . . . . 10 𝑗 βŠ† 𝒫 βˆͺ 𝑗
13 pweq 4617 . . . . . . . . . 10 (𝑏 = βˆͺ 𝑗 β†’ 𝒫 𝑏 = 𝒫 βˆͺ 𝑗)
1412, 13sseqtrrid 4036 . . . . . . . . 9 (𝑏 = βˆͺ 𝑗 β†’ 𝑗 βŠ† 𝒫 𝑏)
15 velpw 4608 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 βŠ† 𝒫 𝑏)
1614, 15sylibr 233 . . . . . . . 8 (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏)
1716a1i 11 . . . . . . 7 (𝑗 ∈ Top β†’ (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏))
1811, 17mprgbir 3069 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏
1910, 18ssexi 5323 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} ∈ V
207, 8, 19fvmpt3i 7004 . . . 4 (𝐡 ∈ V β†’ (TopOnβ€˜π΅) = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
2120eleq2d 2820 . . 3 (𝐡 ∈ V β†’ (𝐽 ∈ (TopOnβ€˜π΅) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗}))
22 unieq 4920 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
2322eqeq2d 2744 . . . 4 (𝑗 = 𝐽 β†’ (𝐡 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝐽))
2423elrab 3684 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
2521, 24bitrdi 287 . 2 (𝐡 ∈ V β†’ (𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽)))
261, 5, 25pm5.21nii 380 1 (𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topon 22413
This theorem is referenced by:  topontop  22415  toponuni  22416  toptopon  22419  toponcom  22430  istps2  22437  tgtopon  22474  distopon  22500  indistopon  22504  fctop  22507  cctop  22509  ppttop  22510  epttop  22512  mretopd  22596  toponmre  22597  resttopon  22665  resttopon2  22672  kgentopon  23042  txtopon  23095  pttopon  23100  xkotopon  23104  qtoptopon  23208  flimtopon  23474  fclstopon  23516  fclsfnflim  23531  utoptopon  23741  qtopt1  32815  neibastop1  35244  onsuctopon  35319  rfcnpre1  43703  cnfex  43712  icccncfext  44603  stoweidlem47  44763
  Copyright terms: Public domain W3C validator