Theorem ntruni 36328

Description: A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
ntruni.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ntruni	⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))

Distinct variable groups:   𝑜,𝐽   𝑜,𝑂   𝑜,𝑋

Proof of Theorem ntruni

Step	Hyp	Ref	Expression
1		elssuni 4937	. . . 4 ⊢ (𝑜 ∈ 𝑂 → 𝑜 ⊆ ∪ 𝑂)
2		sspwuni 5100	. . . . 5 ⊢ (𝑂 ⊆ 𝒫 𝑋 ↔ ∪ 𝑂 ⊆ 𝑋)
3		ntruni.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrss 23063	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋 ∧ 𝑜 ⊆ ∪ 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
5	4	3expia 1122	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋) → (𝑜 ⊆ ∪ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
6	2, 5	sylan2b 594	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜 ⊆ ∪ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
7	1, 6	syl5 34	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜 ∈ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
8	7	ralrimiv 3145	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
9		iunss 5045	. 2 ⊢ (∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂) ↔ ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
10	8, 9	sylibr 234	1 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ∪ ciun 4991  ‘cfv 6561  Topctop 22899  intcnt 23025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029

This theorem is referenced by: (None)