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Theorem istopon 13382
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 13381 . . . . 5 Fun TopOn
2 funrel 5232 . . . . 5 (Fun TopOn → Rel TopOn)
31, 2ax-mp 5 . . . 4 Rel TopOn
4 relelfvdm 5546 . . . 4 ((Rel TopOn ∧ 𝐽 ∈ (TopOn‘𝐵)) → 𝐵 ∈ dom TopOn)
53, 4mpan 424 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ dom TopOn)
65elexd 2750 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
7 uniexg 4438 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
8 eleq1 2240 . . . 4 (𝐵 = 𝐽 → (𝐵 ∈ V ↔ 𝐽 ∈ V))
97, 8syl5ibrcom 157 . . 3 (𝐽 ∈ Top → (𝐵 = 𝐽𝐵 ∈ V))
109imp 124 . 2 ((𝐽 ∈ Top ∧ 𝐵 = 𝐽) → 𝐵 ∈ V)
11 eqeq1 2184 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
1211rabbidv 2726 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
13 df-topon 13380 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
14 vpwex 4178 . . . . . . 7 𝒫 𝑏 ∈ V
1514pwex 4182 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
16 rabss 3232 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
17 pwuni 4191 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
18 pweq 3578 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1917, 18sseqtrrid 3206 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
20 velpw 3582 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏𝑗 ⊆ 𝒫 𝑏)
2119, 20sylibr 134 . . . . . . . 8 (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏)
2221a1i 9 . . . . . . 7 (𝑗 ∈ Top → (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
2316, 22mprgbir 2535 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
2415, 23ssexi 4140 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
2512, 13, 24fvmpt3i 5595 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2625eleq2d 2247 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
27 unieq 3818 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2827eqeq2d 2189 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2928elrab 2893 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
3026, 29bitrdi 196 . 2 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽)))
316, 10, 30pm5.21nii 704 1 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2737  wss 3129  𝒫 cpw 3575   cuni 3809  dom cdm 4625  Rel wrel 4630  Fun wfun 5209  cfv 5215  Topctop 13366  TopOnctopon 13379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-topon 13380
This theorem is referenced by:  topontop  13383  toponuni  13384  toptopon  13387  toponcom  13396  istps2  13402  tgtopon  13437  distopon  13458  epttop  13461  resttopon  13542  resttopon2  13549  txtopon  13633
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