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Theorem tgval 20683
Description: The topology generated by a basis. See also tgval2 20684 and tgval3 20691. (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 3201 . 2 (𝐵𝑉𝐵 ∈ V)
2 uniexg 6915 . . 3 (𝐵𝑉 𝐵 ∈ V)
3 abssexg 4816 . . 3 ( 𝐵 ∈ V → {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V)
4 uniin 4428 . . . . . . 7 (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)
5 sstr 3595 . . . . . . 7 ((𝑥 (𝐵 ∩ 𝒫 𝑥) ∧ (𝐵 ∩ 𝒫 𝑥) ⊆ ( 𝐵 𝒫 𝑥)) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
64, 5mpan2 706 . . . . . 6 (𝑥 (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
7 ssin 3818 . . . . . 6 ((𝑥 𝐵𝑥 𝒫 𝑥) ↔ 𝑥 ⊆ ( 𝐵 𝒫 𝑥))
86, 7sylibr 224 . . . . 5 (𝑥 (𝐵 ∩ 𝒫 𝑥) → (𝑥 𝐵𝑥 𝒫 𝑥))
98ss2abi 3658 . . . 4 {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)}
10 ssexg 4769 . . . 4 (({𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ⊆ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∧ {𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V) → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
119, 10mpan 705 . . 3 ({𝑥 ∣ (𝑥 𝐵𝑥 𝒫 𝑥)} ∈ V → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
122, 3, 113syl 18 . 2 (𝐵𝑉 → {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V)
13 ineq1 3790 . . . . . 6 (𝑦 = 𝐵 → (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1413unieqd 4417 . . . . 5 (𝑦 = 𝐵 (𝑦 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑥))
1514sseq2d 3617 . . . 4 (𝑦 = 𝐵 → (𝑥 (𝑦 ∩ 𝒫 𝑥) ↔ 𝑥 (𝐵 ∩ 𝒫 𝑥)))
1615abbidv 2738 . . 3 (𝑦 = 𝐵 → {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)} = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
17 df-topgen 16036 . . 3 topGen = (𝑦 ∈ V ↦ {𝑥𝑥 (𝑦 ∩ 𝒫 𝑥)})
1816, 17fvmptg 6242 . 2 ((𝐵 ∈ V ∧ {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)} ∈ V) → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
191, 12, 18syl2anc 692 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3189  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407  cfv 5852  topGenctg 16030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-topgen 16036
This theorem is referenced by:  tgval2  20684  eltg  20685  tgdif0  20720
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