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Theorem istopon 12023
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 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))

Proof of Theorem istopon
Dummy variables 𝑏 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funtopon 12022 . . . . 5 Fun TopOn
2 funrel 5098 . . . . 5 (Fun TopOn → Rel TopOn)
31, 2ax-mp 7 . . . 4 Rel TopOn
4 relelfvdm 5407 . . . 4 ((Rel TopOn ∧ 𝐽 ∈ (TopOn‘𝐵)) → 𝐵 ∈ dom TopOn)
53, 4mpan 418 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ dom TopOn)
65elexd 2670 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
7 uniexg 4321 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
8 eleq1 2177 . . . 4 (𝐵 = 𝐽 → (𝐵 ∈ V ↔ 𝐽 ∈ V))
97, 8syl5ibrcom 156 . . 3 (𝐽 ∈ Top → (𝐵 = 𝐽𝐵 ∈ V))
109imp 123 . 2 ((𝐽 ∈ Top ∧ 𝐵 = 𝐽) → 𝐵 ∈ V)
11 eqeq1 2121 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
1211rabbidv 2646 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
13 df-topon 12021 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
14 vpwex 4063 . . . . . . 7 𝒫 𝑏 ∈ V
1514pwex 4067 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
16 rabss 3140 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
17 pwuni 4076 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
18 pweq 3479 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1917, 18sseqtrrid 3114 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
20 selpw 3483 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏𝑗 ⊆ 𝒫 𝑏)
2119, 20sylibr 133 . . . . . . . 8 (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏)
2221a1i 9 . . . . . . 7 (𝑗 ∈ Top → (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
2316, 22mprgbir 2464 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
2415, 23ssexi 4026 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
2512, 13, 24fvmpt3i 5455 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2625eleq2d 2184 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
27 unieq 3711 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2827eqeq2d 2126 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2928elrab 2809 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
3026, 29syl6bb 195 . 2 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽)))
316, 10, 30pm5.21nii 676 1 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  {crab 2394  Vcvv 2657  wss 3037  𝒫 cpw 3476   cuni 3702  dom cdm 4499  Rel wrel 4504  Fun wfun 5075  cfv 5081  Topctop 12007  TopOnctopon 12020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-topon 12021
This theorem is referenced by:  topontop  12024  toponuni  12025  toptopon  12028  toponcom  12037  istps2  12043  tgtopon  12078  distopon  12099  epttop  12102  resttopon  12183  resttopon2  12190  txtopon  12273
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