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Theorem tgval 22802
Description: The topology generated by a basis. See also tgval2 22803 and tgval3 22810. (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 17394 . 2 topGen = (𝑦 ∈ V ↦ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)})
2 ineq1 4198 . . . . 5 (𝑦 = 𝐡 β†’ (𝑦 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 π‘₯))
32unieqd 4913 . . . 4 (𝑦 = 𝐡 β†’ βˆͺ (𝑦 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
43sseq2d 4007 . . 3 (𝑦 = 𝐡 β†’ (π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯) ↔ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)))
54abbidv 2793 . 2 (𝑦 = 𝐡 β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)} = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
6 elex 3485 . 2 (𝐡 ∈ 𝑉 β†’ 𝐡 ∈ V)
7 uniexg 7724 . . 3 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
8 abssexg 5371 . . 3 (βˆͺ 𝐡 ∈ V β†’ {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∈ V)
9 uniin 4926 . . . . . . 7 βˆͺ (𝐡 ∩ 𝒫 π‘₯) βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯)
10 sstr 3983 . . . . . . 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 688 . . . . . 6 (π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯) β†’ π‘₯ βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯))
12 ssin 4223 . . . . . 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 4056 . . . 4 {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} βŠ† {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)}
15 ssexg 5314 . . . 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 687 . . 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 7012 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 395   = wceq 1533   ∈ wcel 2098  {cab 2701  Vcvv 3466   ∩ cin 3940   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900  β€˜cfv 6534  topGenctg 17388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-topgen 17394
This theorem is referenced by:  tgval2  22803  eltg  22804  tgdif0  22839
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