Theorem tgiun 22702

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 5963	. . 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 2732	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	rnmptss 7124	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵)
5		eltg3i 22684	. . 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 2833	1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3948  ∪ cuni 4908  ∪ ciun 4997   ↦ cmpt 5231  ran crn 5677  ‘cfv 6543  topGenctg 17387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-topgen 17393

This theorem is referenced by:  txbasval  23330