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Mirrors > Home > MPE Home > Th. List > ntrdif | Structured version Visualization version GIF version |
Description: An interior of a complement is the complement of the closure. This set is also known as the exterior of 𝐴. (Contributed by Jeff Hankins, 31-Aug-2009.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ntrdif | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4107 | . . . 4 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋 | |
2 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | ntrval2 21653 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) |
4 | 1, 3 | mpan2 689 | . . 3 ⊢ (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) |
6 | simpr 487 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
7 | dfss4 4234 | . . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | |
8 | 6, 7 | sylib 220 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) |
9 | 8 | fveq2d 6668 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((cls‘𝐽)‘𝐴)) |
10 | 9 | difeq2d 4098 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))) = (𝑋 ∖ ((cls‘𝐽)‘𝐴))) |
11 | 5, 10 | eqtrd 2856 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝐴)) = (𝑋 ∖ ((cls‘𝐽)‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ⊆ wss 3935 ∪ cuni 4831 ‘cfv 6349 Topctop 21495 intcnt 21619 clsccl 21620 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-top 21496 df-cld 21621 df-ntr 21622 df-cls 21623 |
This theorem is referenced by: clsun 33671 |
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