Theorem iunopn 12158

Description: The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
iunopn	⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunopn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 3840	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 275	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		uniiunlem 3180	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽))
4	3	ibi 175	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5		uniopn 12157	. . 3 ⊢ ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽)
6	4, 5	sylan2 284	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽)
7	2, 6	eqeltrd 2214	1 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2123  ∀wral 2414  ∃wrex 2415   ⊆ wss 3066  ∪ cuni 3731  ∪ ciun 3808  Topctop 12153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-uni 3732  df-iun 3810  df-top 12154

This theorem is referenced by:  tgcn  12366