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Theorem istopon 13483
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 funtopon 13482 . . . . 5 Fun TopOn
2 funrel 5233 . . . . 5 (Fun TopOn β†’ Rel TopOn)
31, 2ax-mp 5 . . . 4 Rel TopOn
4 relelfvdm 5547 . . . 4 ((Rel TopOn ∧ 𝐽 ∈ (TopOnβ€˜π΅)) β†’ 𝐡 ∈ dom TopOn)
53, 4mpan 424 . . 3 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 ∈ dom TopOn)
65elexd 2750 . 2 (𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 ∈ V)
7 uniexg 4439 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
8 eleq1 2240 . . . 4 (𝐡 = βˆͺ 𝐽 β†’ (𝐡 ∈ V ↔ βˆͺ 𝐽 ∈ V))
97, 8syl5ibrcom 157 . . 3 (𝐽 ∈ Top β†’ (𝐡 = βˆͺ 𝐽 β†’ 𝐡 ∈ V))
109imp 124 . 2 ((𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽) β†’ 𝐡 ∈ V)
11 eqeq1 2184 . . . . . 6 (𝑏 = 𝐡 β†’ (𝑏 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝑗))
1211rabbidv 2726 . . . . 5 (𝑏 = 𝐡 β†’ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
13 df-topon 13481 . . . . 5 TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗})
14 vpwex 4179 . . . . . . 7 𝒫 𝑏 ∈ V
1514pwex 4183 . . . . . 6 𝒫 𝒫 𝑏 ∈ V
16 rabss 3232 . . . . . . 7 ({𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏 ↔ βˆ€π‘— ∈ Top (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏))
17 pwuni 4192 . . . . . . . . . 10 𝑗 βŠ† 𝒫 βˆͺ 𝑗
18 pweq 3578 . . . . . . . . . 10 (𝑏 = βˆͺ 𝑗 β†’ 𝒫 𝑏 = 𝒫 βˆͺ 𝑗)
1917, 18sseqtrrid 3206 . . . . . . . . 9 (𝑏 = βˆͺ 𝑗 β†’ 𝑗 βŠ† 𝒫 𝑏)
20 velpw 3582 . . . . . . . . 9 (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 βŠ† 𝒫 𝑏)
2119, 20sylibr 134 . . . . . . . 8 (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏)
2221a1i 9 . . . . . . 7 (𝑗 ∈ Top β†’ (𝑏 = βˆͺ 𝑗 β†’ 𝑗 ∈ 𝒫 𝒫 𝑏))
2316, 22mprgbir 2535 . . . . . 6 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} βŠ† 𝒫 𝒫 𝑏
2415, 23ssexi 4141 . . . . 5 {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗} ∈ V
2512, 13, 24fvmpt3i 5596 . . . 4 (𝐡 ∈ V β†’ (TopOnβ€˜π΅) = {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗})
2625eleq2d 2247 . . 3 (𝐡 ∈ V β†’ (𝐽 ∈ (TopOnβ€˜π΅) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗}))
27 unieq 3818 . . . . 5 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
2827eqeq2d 2189 . . . 4 (𝑗 = 𝐽 β†’ (𝐡 = βˆͺ 𝑗 ↔ 𝐡 = βˆͺ 𝐽))
2928elrab 2893 . . 3 (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐡 = βˆͺ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
3026, 29bitrdi 196 . 2 (𝐡 ∈ V β†’ (𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽)))
316, 10, 30pm5.21nii 704 1 (𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {crab 2459  Vcvv 2737   βŠ† wss 3129  π’« cpw 3575  βˆͺ cuni 3809  dom cdm 4626  Rel wrel 4631  Fun wfun 5210  β€˜cfv 5216  Topctop 13467  TopOnctopon 13480
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-topon 13481
This theorem is referenced by:  topontop  13484  toponuni  13485  toptopon  13488  toponcom  13497  istps2  13503  tgtopon  13536  distopon  13557  epttop  13560  resttopon  13641  resttopon2  13648  txtopon  13732
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