Theorem iunopn 14559

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 3968	. . 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 3286	. . . 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 14558	. . 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 2283	1 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486   ⊆ wss 3170  ∪ cuni 3859  ∪ ciun 3936  Topctop 14554

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4173

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623  df-uni 3860  df-iun 3938  df-top 14555

This theorem is referenced by:  tgcn  14765