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Theorem tgval 22871
Description: The topology generated by a basis. See also tgval2 22872 and tgval3 22879. (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 17425 . 2 topGen = (𝑦 ∈ V ↦ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)})
2 ineq1 4205 . . . . 5 (𝑦 = 𝐡 β†’ (𝑦 ∩ 𝒫 π‘₯) = (𝐡 ∩ 𝒫 π‘₯))
32unieqd 4921 . . . 4 (𝑦 = 𝐡 β†’ βˆͺ (𝑦 ∩ 𝒫 π‘₯) = βˆͺ (𝐡 ∩ 𝒫 π‘₯))
43sseq2d 4012 . . 3 (𝑦 = 𝐡 β†’ (π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯) ↔ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)))
54abbidv 2797 . 2 (𝑦 = 𝐡 β†’ {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝑦 ∩ 𝒫 π‘₯)} = {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)})
6 elex 3490 . 2 (𝐡 ∈ 𝑉 β†’ 𝐡 ∈ V)
7 uniexg 7745 . . 3 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
8 abssexg 5382 . . 3 (βˆͺ 𝐡 ∈ V β†’ {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)} ∈ V)
9 uniin 4934 . . . . . . 7 βˆͺ (𝐡 ∩ 𝒫 π‘₯) βŠ† (βˆͺ 𝐡 ∩ βˆͺ 𝒫 π‘₯)
10 sstr 3988 . . . . . . 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 4231 . . . . . 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 4061 . . . 4 {π‘₯ ∣ π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯)} βŠ† {π‘₯ ∣ (π‘₯ βŠ† βˆͺ 𝐡 ∧ π‘₯ βŠ† βˆͺ 𝒫 π‘₯)}
15 ssexg 5323 . . . 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 7028 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 1534   ∈ wcel 2099  {cab 2705  Vcvv 3471   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4603  βˆͺ cuni 4908  β€˜cfv 6548  topGenctg 17419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topgen 17425
This theorem is referenced by:  tgval2  22872  eltg  22873  tgdif0  22908
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