Theorem tgiun 23309

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 5954	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶))
2	1	adantl 487	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶))
3		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	rnmptss 7131	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵)
5		eltg3i 23291	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵))
6	4, 5	sylan2 605	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵))
7	2, 6	eqeltrd 2863	1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ⊆ wss 3899  ∪ cuni 4869  ∪ ciun 4953   ↦ cmpt 5188  ran crn 5655  ‘cfv 6546  topGenctg 17620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-topgen 17626

This theorem is used by:  txbasval  23937