Theorem tgiun 22129

Description: The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
tgiun	⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))

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

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem tgiun

Step	Hyp	Ref	Expression
1		dfiun3g 5873	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶))
2	1	adantl 482	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶))
3		eqid 2738	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	rnmptss 6996	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵)
5		eltg3i 22111	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵))
6	4, 5	sylan2 593	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵))
7	2, 6	eqeltrd 2839	1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ∪ cuni 4839  ∪ ciun 4924   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  topGenctg 17148

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-topgen 17154

This theorem is referenced by:  txbasval  22757