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Theorem istopon 12350
 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 12349 . . . . 5 Fun TopOn
2 funrel 5180 . . . . 5 (Fun TopOn → Rel TopOn)
31, 2ax-mp 5 . . . 4 Rel TopOn
4 relelfvdm 5493 . . . 4 ((Rel TopOn ∧ 𝐽 ∈ (TopOn‘𝐵)) → 𝐵 ∈ dom TopOn)
53, 4mpan 421 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ dom TopOn)
65elexd 2722 . 2 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
7 uniexg 4394 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
8 eleq1 2217 . . . 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 2161 . . . . . 6 (𝑏 = 𝐵 → (𝑏 = 𝑗𝐵 = 𝑗))
1211rabbidv 2698 . . . . 5 (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
13 df-topon 12348 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
14 vpwex 4135 . . . . . . 7 𝒫 𝑏 ∈ V
1514pwex 4139 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
16 rabss 3201 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = 𝑗𝑗 ∈ 𝒫 𝒫 𝑏))
17 pwuni 4148 . . . . . . . . . 10 𝑗 ⊆ 𝒫 𝑗
18 pweq 3542 . . . . . . . . . 10 (𝑏 = 𝑗 → 𝒫 𝑏 = 𝒫 𝑗)
1917, 18sseqtrrid 3175 . . . . . . . . 9 (𝑏 = 𝑗𝑗 ⊆ 𝒫 𝑏)
20 velpw 3546 . . . . . . . . 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 2512 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ⊆ 𝒫 𝒫 𝑏
2415, 23ssexi 4098 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = 𝑗} ∈ V
2512, 13, 24fvmpt3i 5541 . . . 4 (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = 𝑗})
2625eleq2d 2224 . . 3 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗}))
27 unieq 3777 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2827eqeq2d 2166 . . . 4 (𝑗 = 𝐽 → (𝐵 = 𝑗𝐵 = 𝐽))
2928elrab 2864 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
3026, 29bitrdi 195 . 2 (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽)))
316, 10, 30pm5.21nii 694 1 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2125  {crab 2436  Vcvv 2709   ⊆ wss 3098  𝒫 cpw 3539  ∪ cuni 3768  dom cdm 4579  Rel wrel 4584  Fun wfun 5157  ‘cfv 5163  Topctop 12334  TopOnctopon 12347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-topon 12348 This theorem is referenced by:  topontop  12351  toponuni  12352  toptopon  12355  toponcom  12364  istps2  12370  tgtopon  12405  distopon  12426  epttop  12429  resttopon  12510  resttopon2  12517  txtopon  12601
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