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Theorem tgval 12843
Description: The topology generated by a basis. See also tgval2 12845 and tgval3 12852. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem tgval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 2741 . 2 (𝐵𝑉𝐵 ∈ V)
2 uniexg 4424 . . 3 (𝐵𝑉 𝐵 ∈ V)
3 abssexg 4168 . . 3 ( 𝐵 ∈ V → {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V)
4 uniin 3816 . . . . . . 7 (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)
5 sstr 3155 . . . . . . 7 ((𝑥 (𝐵 ∩ 𝒫 𝑥) ∧ (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
64, 5mpan2 423 . . . . . 6 (𝑥 (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
7 ssin 3349 . . . . . 6 ((𝑥 𝐵𝑥 𝒫 𝑥) ↔ 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
86, 7sylibr 133 . . . . 5 (𝑥 (𝐵 ∩ 𝒫 𝑥) → (𝑥 𝐵𝑥 𝒫 𝑥))
98ss2abi 3219 . . . 4 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)}
10 ssexg 4128 . . . 4 (({𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V) → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
119, 10mpan 422 . . 3 ({𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
122, 3, 113syl 17 . 2 (𝐵𝑉 → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
13 ineq1 3321 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1413unieqd 3807 . . . . 5 (𝑦 = 𝐵 (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1514sseq2d 3177 . . . 4 (𝑦 = 𝐵 → (𝑥 (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
1615abbidv 2288 . . 3 (𝑦 = 𝐵 → {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
17 df-topgen 12600 . . 3 topGen = (𝑦 ∈ V ↦ {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)})
1816, 17fvmptg 5572 . 2 ((𝐵 ∈ V ∧ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V) → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
191, 12, 18syl2anc 409 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  {cab 2156  Vcvv 2730  cin 3120  wss 3121  𝒫 cpw 3566   cuni 3796  cfv 5198  topGenctg 12594
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-topgen 12600
This theorem is referenced by:  tgvalex  12844  tgval2  12845  eltg  12846
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