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Theorem tgval3 22466
Description: Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 22458 and tgval2 22459. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
tgval3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)})
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝑉,𝑦

Proof of Theorem tgval3
StepHypRef Expression
1 eltg3 22465 . 2 (𝐡 ∈ 𝑉 β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
21eqabdv 2868 1 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {π‘₯ ∣ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  topGenctg 17383
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-topgen 17389
This theorem is referenced by: (None)
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