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Theorem tgval3 20687
Description: Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 20679 and tgval2 20680. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
tgval3 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑉,𝑦

Proof of Theorem tgval3
StepHypRef Expression
1 eltg3 20686 . 2 (𝐵𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦𝐵𝑥 = 𝑦)))
21abbi2dv 2739 1 (𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wss 3559   cuni 4407  cfv 5852  topGenctg 16026
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-ne 2791  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 16032
This theorem is referenced by: (None)
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