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Theorem tgval 22321
Description: The topology generated by a basis. See also tgval2 22322 and tgval3 22329. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
tgval (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑉

Proof of Theorem tgval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-topgen 17330 . 2 topGen = (𝑦 ∈ V ↦ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)})
2 ineq1 4166 . . . . 5 (𝑦 = 𝐡 β†’ (𝑦 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 π‘₯))
32unieqd 4880 . . . 4 (𝑦 = 𝐡 β†’ βˆͺ (𝑦 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
43sseq2d 3977 . . 3 (𝑦 = 𝐡 β†’ (π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯) ↔ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)))
54abbidv 2802 . 2 (𝑦 = 𝐡 β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)} = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
6 elex 3462 . 2 (𝐡 ∈ 𝑉 β†’ 𝐡 ∈ V)
7 uniexg 7678 . . 3 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
8 abssexg 5338 . . 3 (βˆͺ 𝐡 ∈ V β†’ {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∈ V)
9 uniin 4893 . . . . . . 7 βˆͺ (𝐡 ∩ 𝒫 π‘₯) βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯)
10 sstr 3953 . . . . . . 7 ((π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) ∧ βˆͺ (𝐡 ∩ 𝒫 π‘₯) βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯)) β†’ π‘₯ βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯))
119, 10mpan2 690 . . . . . 6 (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) β†’ π‘₯ βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯))
12 ssin 4191 . . . . . 6 ((π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯) ↔ π‘₯ βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯))
1311, 12sylibr 233 . . . . 5 (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) β†’ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯))
1413ss2abi 4024 . . . 4 {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} βŠ† {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)}
15 ssexg 5281 . . . 4 (({π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} βŠ† {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∧ {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∈ V) β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ∈ V)
1614, 15mpan 689 . . 3 ({π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∈ V β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ∈ V)
177, 8, 163syl 18 . 2 (𝐡 ∈ 𝑉 β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} ∈ V)
181, 5, 6, 17fvmptd3 6972 1 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  β€˜cfv 6497  topGenctg 17324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-topgen 17330
This theorem is referenced by:  tgval2  22322  eltg  22323  tgdif0  22358
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