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Theorem ntruni 36310
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) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
Distinct variable groups:   𝑜,𝐽   𝑜,𝑂   𝑜,𝑋

Proof of Theorem ntruni
StepHypRef Expression
1 elssuni 4942 . . . 4 (𝑜𝑂𝑜 𝑂)
2 sspwuni 5105 . . . . 5 (𝑂 ⊆ 𝒫 𝑋 𝑂𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = 𝐽
43ntrss 23079 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑂𝑋𝑜 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
543expia 1120 . . . . 5 ((𝐽 ∈ Top ∧ 𝑂𝑋) → (𝑜 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂)))
62, 5sylan2b 594 . . . 4 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂)))
71, 6syl5 34 . . 3 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → (𝑜𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂)))
87ralrimiv 3143 . 2 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
9 iunss 5050 . 2 ( 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂) ↔ ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
108, 9sylibr 234 1 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  𝒫 cpw 4605   cuni 4912   ciun 4996  cfv 6563  Topctop 22915  intcnt 23041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045
This theorem is referenced by: (None)
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