Theorem iunopn 14238

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 3948	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 277	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		uniiunlem 3272	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽))
4	3	ibi 176	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽)
5		uniopn 14237	. . 3 ⊢ ((𝐽 ∈ Top ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽)
6	4, 5	sylan2 286	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽)
7	2, 6	eqeltrd 2273	1 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476   ⊆ wss 3157  ∪ cuni 3839  ∪ ciun 3916  Topctop 14233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840  df-iun 3918  df-top 14234

This theorem is referenced by:  tgcn  14444