Theorem ntruni 36322

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 4904	. . . 4 ⊢ (𝑜 ∈ 𝑂 → 𝑜 ⊆ ∪ 𝑂)
2		sspwuni 5067	. . . . 5 ⊢ (𝑂 ⊆ 𝒫 𝑋 ↔ ∪ 𝑂 ⊆ 𝑋)
3		ntruni.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrss 22949	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋 ∧ 𝑜 ⊆ ∪ 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
5	4	3expia 1121	. . . . 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 3125	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
9		iunss 5012	. 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 2109  ∀wral 3045   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ∪ ciun 4958  ‘cfv 6514  Topctop 22787  intcnt 22911

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-top 22788  df-cld 22913  df-ntr 22914  df-cls 22915

This theorem is referenced by: (None)