Theorem tgval3 23131

Description: Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 23123 and tgval2 23124. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
tgval3	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)})

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑉,𝑦

Proof of Theorem tgval3

Step	Hyp	Ref	Expression
1		eltg3 23130	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
2	1	eqabdv 2895	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740   ⊆ wss 3904  ∪ cuni 4871  ‘cfv 6536  topGenctg 17496

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-topgen 17502

This theorem is used by: (None)