Theorem ntruni 37207

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 4898	. . . 4 ⊢ (𝑜 ∈ 𝑂 → 𝑜 ⊆ ∪ 𝑂)
2		sspwuni 5059	. . . . 5 ⊢ (𝑂 ⊆ 𝒫 𝑋 ↔ ∪ 𝑂 ⊆ 𝑋)
3		ntruni.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	ntrss 23378	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋 ∧ 𝑜 ⊆ ∪ 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
5	4	3expia 1139	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋) → (𝑜 ⊆ ∪ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
6	2, 5	sylan2b 606	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜 ⊆ ∪ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
7	1, 6	syl5 35	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜 ∈ 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂)))
8	7	ralrimiv 3153	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
9		iunss 5002	. 2 ⊢ (∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂) ↔ ∀𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))
10	8, 9	sylibr 237	1 ⊢ ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∪ 𝑜 ∈ 𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘∪ 𝑂))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866  ∪ ciun 4950  ‘cfv 6538  Topctop 23216  intcnt 23340

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23217  df-cld 23342  df-ntr 23343  df-cls 23344

This theorem is used by: (None)