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Theorem clsval2 22545
Description: Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
clsval2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
Proof of Theorem clsval2
Dummy variables π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 3433 . . . . . 6 {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = βˆͺ 𝐽
32cldopn 22526 . . . . . . . . . . . 12 (𝑧 ∈ (Clsdβ€˜π½) β†’ (𝑋 βˆ– 𝑧) ∈ 𝐽)
43ad2antrl 726 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ 𝐽)
5 sscon 4137 . . . . . . . . . . . . 13 (𝑆 βŠ† 𝑧 β†’ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆))
65ad2antll 727 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆))
72topopn 22399 . . . . . . . . . . . . . 14 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
8 difexg 5326 . . . . . . . . . . . . . 14 (𝑋 ∈ 𝐽 β†’ (𝑋 βˆ– 𝑧) ∈ V)
9 elpwg 4604 . . . . . . . . . . . . . 14 ((𝑋 βˆ– 𝑧) ∈ V β†’ ((𝑋 βˆ– 𝑧) ∈ 𝒫 (𝑋 βˆ– 𝑆) ↔ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆)))
107, 8, 93syl 18 . . . . . . . . . . . . 13 (𝐽 ∈ Top β†’ ((𝑋 βˆ– 𝑧) ∈ 𝒫 (𝑋 βˆ– 𝑆) ↔ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆)))
1110ad2antrr 724 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ ((𝑋 βˆ– 𝑧) ∈ 𝒫 (𝑋 βˆ– 𝑆) ↔ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆)))
126, 11mpbird 256 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ 𝒫 (𝑋 βˆ– 𝑆))
134, 12elind 4193 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
142cldss 22524 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsdβ€˜π½) β†’ 𝑧 βŠ† 𝑋)
1514ad2antrl 726 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ 𝑧 βŠ† 𝑋)
16 dfss4 4257 . . . . . . . . . . . 12 (𝑧 βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– 𝑧)) = 𝑧)
1715, 16sylib 217 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– (𝑋 βˆ– 𝑧)) = 𝑧)
1817eqcomd 2738 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ 𝑧 = (𝑋 βˆ– (𝑋 βˆ– 𝑧)))
19 difeq2 4115 . . . . . . . . . . 11 (π‘₯ = (𝑋 βˆ– 𝑧) β†’ (𝑋 βˆ– π‘₯) = (𝑋 βˆ– (𝑋 βˆ– 𝑧)))
2019rspceeqv 3632 . . . . . . . . . 10 (((𝑋 βˆ– 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) ∧ 𝑧 = (𝑋 βˆ– (𝑋 βˆ– 𝑧))) β†’ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯))
2113, 18, 20syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯))
2221ex 413 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧) β†’ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)))
23 simpl 483 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
24 elinel1 4194 . . . . . . . . . . . 12 (π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ π‘₯ ∈ 𝐽)
252opncld 22528 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽) β†’ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½))
2623, 24, 25syl2an 596 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½))
27 elinel2 4195 . . . . . . . . . . . . . 14 (π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ π‘₯ ∈ 𝒫 (𝑋 βˆ– 𝑆))
2827adantl 482 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ ∈ 𝒫 (𝑋 βˆ– 𝑆))
2928elpwid 4610 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ βŠ† (𝑋 βˆ– 𝑆))
3029difss2d 4133 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ βŠ† 𝑋)
31 simplr 767 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ 𝑆 βŠ† 𝑋)
32 ssconb 4136 . . . . . . . . . . . . 13 ((π‘₯ βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ βŠ† (𝑋 βˆ– 𝑆) ↔ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
3330, 31, 32syl2anc 584 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ (π‘₯ βŠ† (𝑋 βˆ– 𝑆) ↔ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
3429, 33mpbid 231 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ 𝑆 βŠ† (𝑋 βˆ– π‘₯))
3526, 34jca 512 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ ((𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
36 eleq1 2821 . . . . . . . . . . 11 (𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑧 ∈ (Clsdβ€˜π½) ↔ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½)))
37 sseq2 4007 . . . . . . . . . . 11 (𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑆 βŠ† 𝑧 ↔ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
3836, 37anbi12d 631 . . . . . . . . . 10 (𝑧 = (𝑋 βˆ– π‘₯) β†’ ((𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧) ↔ ((𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† (𝑋 βˆ– π‘₯))))
3935, 38syl5ibrcom 246 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ (𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)))
4039rexlimdva 3155 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)))
4122, 40impbid 211 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧) ↔ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)))
4241abbidv 2801 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ {𝑧 ∣ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)} = {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
431, 42eqtrid 2784 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
4443inteqd 4954 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ∩ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = ∩ {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
45 difexg 5326 . . . . . . 7 (𝑋 ∈ 𝐽 β†’ (𝑋 βˆ– π‘₯) ∈ V)
4645ralrimivw 3150 . . . . . 6 (𝑋 ∈ 𝐽 β†’ βˆ€π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) ∈ V)
47 dfiin2g 5034 . . . . . 6 (βˆ€π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) ∈ V β†’ ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) = ∩ {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
487, 46, 473syl 18 . . . . 5 (𝐽 ∈ Top β†’ ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) = ∩ {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
4948adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) = ∩ {𝑧 ∣ βˆƒπ‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))𝑧 = (𝑋 βˆ– π‘₯)})
5044, 49eqtr4d 2775 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ∩ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯))
512clsval 22532 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = ∩ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧})
52 uniiun 5060 . . . . . 6 βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) = βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯
5352difeq2i 4118 . . . . 5 (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) = (𝑋 βˆ– βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯)
5453a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) = (𝑋 βˆ– βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯))
55 0opn 22397 . . . . . . 7 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
5655adantr 481 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ 𝐽)
57 0elpw 5353 . . . . . . 7 βˆ… ∈ 𝒫 (𝑋 βˆ– 𝑆)
5857a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ 𝒫 (𝑋 βˆ– 𝑆))
5956, 58elind 4193 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
60 ne0i 4333 . . . . 5 (βˆ… ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β‰  βˆ…)
61 iindif2 5079 . . . . 5 ((𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β‰  βˆ… β†’ ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) = (𝑋 βˆ– βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯))
6259, 60, 613syl 18 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) = (𝑋 βˆ– βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯))
6354, 62eqtr4d 2775 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) = ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯))
6450, 51, 633eqtr4d 2782 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))))
65 difssd 4131 . . . 4 (𝑆 βŠ† 𝑋 β†’ (𝑋 βˆ– 𝑆) βŠ† 𝑋)
662ntrval 22531 . . . 4 ((𝐽 ∈ Top ∧ (𝑋 βˆ– 𝑆) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
6765, 66sylan2 593 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
6867difeq2d 4121 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆))) = (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))))
6964, 68eqtr4d 2775 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βˆͺ cuni 4907  βˆ© cint 4949  βˆͺ ciun 4996  βˆ© ciin 4997  β€˜cfv 6540  Topctop 22386  Clsdccld 22511  intcnt 22512  clsccl 22513
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-cld 22514  df-ntr 22515  df-cls 22516
This theorem is referenced by:  ntrval2  22546  clsdif  22548  cmclsopn  22557  bcth3  24839
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