Theorem ntruni 36525

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 4882	. . . 4 ⊢ (𝑜 ∈ 𝑂 → 𝑜 ⊆ ∪ 𝑂)
2		sspwuni 5043	. . . . 5 ⊢ (𝑂 ⊆ 𝒫 𝑋 ↔ ∪ 𝑂 ⊆ 𝑋)
3		ntruni.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrss 23030	. . . . . 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 595	. . . 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 3129	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
9		iunss 4988	. 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 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ∪ ciun 4934  ‘cfv 6492  Topctop 22868  intcnt 22992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22869  df-cld 22994  df-ntr 22995  df-cls 22996

This theorem is referenced by: (None)