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Theorem istopon 22277
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 6885 . 2 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 ∈ V)
2 uniexg 7682 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
3 eleq1 2826 . . . 4 (𝐡 = βˆͺ 𝐽 β†’ (𝐡 ∈ V ↔ βˆͺ 𝐽 ∈ V))
42, 3syl5ibrcom 247 . . 3 (𝐽 ∈ Top β†’ (𝐡 = βˆͺ 𝐽 β†’ 𝐡 ∈ V))
54imp 408 . 2 ((𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽) β†’ 𝐡 ∈ V)
6 eqeq1 2741 . . . . . 6 (𝑏 = 𝐡 β†’ (𝑏 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝑗))
76rabbidv 3418 . . . . 5 (𝑏 = 𝐡 β†’ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
8 df-topon 22276 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗})
9 vpwex 5337 . . . . . . 7 𝒫 𝑏 ∈ V
109pwex 5340 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
11 rabss 4034 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏 ↔ βˆ€π‘— ∈ Top (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏))
12 pwuni 4911 . . . . . . . . . 10 𝑗 βŠ† 𝒫 βˆͺ 𝑗
13 pweq 4579 . . . . . . . . . 10 (𝑏 = βˆͺ 𝑗 β†’ 𝒫 𝑏 = 𝒫 βˆͺ 𝑗)
1412, 13sseqtrrid 4002 . . . . . . . . 9 (𝑏 = βˆͺ 𝑗 β†’ 𝑗 βŠ† 𝒫 𝑏)
15 velpw 4570 . . . . . . . . 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 3072 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏
1910, 18ssexi 5284 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} ∈ V
207, 8, 19fvmpt3i 6958 . . . 4 (𝐡 ∈ V β†’ (TopOnβ€˜π΅) = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
2120eleq2d 2824 . . 3 (𝐡 ∈ V β†’ (𝐽 ∈ (TopOnβ€˜π΅) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗}))
22 unieq 4881 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
2322eqeq2d 2748 . . . 4 (𝑗 = 𝐽 β†’ (𝐡 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝐽))
2423elrab 3650 . . 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 3410  Vcvv 3448   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  β€˜cfv 6501  Topctop 22258  TopOnctopon 22275
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-topon 22276
This theorem is referenced by:  topontop  22278  toponuni  22279  toptopon  22282  toponcom  22293  istps2  22300  tgtopon  22337  distopon  22363  indistopon  22367  fctop  22370  cctop  22372  ppttop  22373  epttop  22375  mretopd  22459  toponmre  22460  resttopon  22528  resttopon2  22535  kgentopon  22905  txtopon  22958  pttopon  22963  xkotopon  22967  qtoptopon  23071  flimtopon  23337  fclstopon  23379  fclsfnflim  23394  utoptopon  23604  qtopt1  32456  neibastop1  34860  onsuctopon  34935  rfcnpre1  43298  cnfex  43307  icccncfext  44202  stoweidlem47  44362
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