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Theorem ntruni 33785
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 4830 . . . 4 (𝑜𝑂𝑜 𝑂)
2 sspwuni 4985 . . . . 5 (𝑂 ⊆ 𝒫 𝑋 𝑂𝑋)
3 ntruni.1 . . . . . . 7 𝑋 = 𝐽
43ntrss 21660 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑂𝑋𝑜 𝑂) → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
543expia 1118 . . . . 5 ((𝐽 ∈ Top ∧ 𝑂𝑋) → (𝑜 𝑂 → ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂)))
62, 5sylan2b 596 . . . 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 3148 . 2 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
9 iunss 4932 . 2 ( 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂) ↔ ∀𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
108, 9sylibr 237 1 ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wral 3106  wss 3881  𝒫 cpw 4497   cuni 4800   ciun 4881  cfv 6324  Topctop 21498  intcnt 21622
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626
This theorem is referenced by: (None)
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