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Theorem ntruni 31985
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 4435 . . . 4 (𝑜𝑂𝑜 𝑂)
2 sspwuni 4579 . . . . 5 (𝑂 ⊆ 𝒫 𝑋 𝑂𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = 𝐽
43ntrss 20772 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑂𝑋𝑜 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
543expia 1264 . . . . 5 ((𝐽 ∈ Top ∧ 𝑂𝑋) → (𝑜 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂)))
62, 5sylan2b 492 . . . 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 2959 . 2 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
9 iunss 4529 . 2 ( 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂) ↔ ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
108, 9sylibr 224 1 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3556  𝒫 cpw 4132   cuni 4404   ciun 4487  cfv 5849  Topctop 20620  intcnt 20734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-top 20621  df-cld 20736  df-ntr 20737  df-cls 20738
This theorem is referenced by: (None)
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