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Theorem clsval2 22554
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 3434 . . . . . 6 {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)}
2 clscld.1 . . . . . . . . . . . . 13 𝑋 = βˆͺ 𝐽
32cldopn 22535 . . . . . . . . . . . 12 (𝑧 ∈ (Clsdβ€˜π½) β†’ (𝑋 βˆ– 𝑧) ∈ 𝐽)
43ad2antrl 727 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ 𝐽)
5 sscon 4139 . . . . . . . . . . . . 13 (𝑆 βŠ† 𝑧 β†’ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆))
65ad2antll 728 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) βŠ† (𝑋 βˆ– 𝑆))
72topopn 22408 . . . . . . . . . . . . . 14 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
8 difexg 5328 . . . . . . . . . . . . . 14 (𝑋 ∈ 𝐽 β†’ (𝑋 βˆ– 𝑧) ∈ V)
9 elpwg 4606 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . 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 257 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ 𝒫 (𝑋 βˆ– 𝑆))
134, 12elind 4195 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ (𝑋 βˆ– 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
142cldss 22533 . . . . . . . . . . . . 13 (𝑧 ∈ (Clsdβ€˜π½) β†’ 𝑧 βŠ† 𝑋)
1514ad2antrl 727 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ 𝑧 βŠ† 𝑋)
16 dfss4 4259 . . . . . . . . . . . 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 2739 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑧 ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† 𝑧)) β†’ 𝑧 = (𝑋 βˆ– (𝑋 βˆ– 𝑧)))
19 difeq2 4117 . . . . . . . . . . 11 (π‘₯ = (𝑋 βˆ– 𝑧) β†’ (𝑋 βˆ– π‘₯) = (𝑋 βˆ– (𝑋 βˆ– 𝑧)))
2019rspceeqv 3634 . . . . . . . . . 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 585 . . . . . . . . 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 414 . . . . . . . 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 484 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ 𝐽 ∈ Top)
24 elinel1 4196 . . . . . . . . . . . 12 (π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ π‘₯ ∈ 𝐽)
252opncld 22537 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽) β†’ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½))
2623, 24, 25syl2an 597 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½))
27 elinel2 4197 . . . . . . . . . . . . . 14 (π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ π‘₯ ∈ 𝒫 (𝑋 βˆ– 𝑆))
2827adantl 483 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ ∈ 𝒫 (𝑋 βˆ– 𝑆))
2928elpwid 4612 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ βŠ† (𝑋 βˆ– 𝑆))
3029difss2d 4135 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ π‘₯ βŠ† 𝑋)
31 simplr 768 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))) β†’ 𝑆 βŠ† 𝑋)
32 ssconb 4138 . . . . . . . . . . . . 13 ((π‘₯ βŠ† 𝑋 ∧ 𝑆 βŠ† 𝑋) β†’ (π‘₯ βŠ† (𝑋 βˆ– 𝑆) ↔ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
3330, 31, 32syl2anc 585 . . . . . . . . . . . 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 513 . . . . . . . . . 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 2822 . . . . . . . . . . 11 (𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑧 ∈ (Clsdβ€˜π½) ↔ (𝑋 βˆ– π‘₯) ∈ (Clsdβ€˜π½)))
37 sseq2 4009 . . . . . . . . . . 11 (𝑧 = (𝑋 βˆ– π‘₯) β†’ (𝑆 βŠ† 𝑧 ↔ 𝑆 βŠ† (𝑋 βˆ– π‘₯)))
3836, 37anbi12d 632 . . . . . . . . . 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 3156 . . . . . . . 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 2802 . . . . . 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 2785 . . . . 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 4956 . . . 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 5328 . . . . . . 7 (𝑋 ∈ 𝐽 β†’ (𝑋 βˆ– π‘₯) ∈ V)
4645ralrimivw 3151 . . . . . 6 (𝑋 ∈ 𝐽 β†’ βˆ€π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯) ∈ V)
47 dfiin2g 5036 . . . . . 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 482 . . . 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 2776 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ∩ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧} = ∩ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))(𝑋 βˆ– π‘₯))
512clsval 22541 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = ∩ {𝑧 ∈ (Clsdβ€˜π½) ∣ 𝑆 βŠ† 𝑧})
52 uniiun 5062 . . . . . 6 βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) = βˆͺ π‘₯ ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))π‘₯
5352difeq2i 4120 . . . . 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 22406 . . . . . . 7 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
5655adantr 482 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ 𝐽)
57 0elpw 5355 . . . . . . 7 βˆ… ∈ 𝒫 (𝑋 βˆ– 𝑆)
5857a1i 11 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ 𝒫 (𝑋 βˆ– 𝑆))
5956, 58elind 4195 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ βˆ… ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
60 ne0i 4335 . . . . 5 (βˆ… ∈ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β†’ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)) β‰  βˆ…)
61 iindif2 5081 . . . . 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 2776 . . 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 2783 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑋 βˆ– βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆))))
65 difssd 4133 . . . 4 (𝑆 βŠ† 𝑋 β†’ (𝑋 βˆ– 𝑆) βŠ† 𝑋)
662ntrval 22540 . . . 4 ((𝐽 ∈ Top ∧ (𝑋 βˆ– 𝑆) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
6765, 66sylan2 594 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = βˆͺ (𝐽 ∩ 𝒫 (𝑋 βˆ– 𝑆)))
6867difeq2d 4123 . 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 2776 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  βˆ© cint 4951  βˆͺ ciun 4998  βˆ© ciin 4999  β€˜cfv 6544  Topctop 22395  Clsdccld 22520  intcnt 22521  clsccl 22522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525
This theorem is referenced by:  ntrval2  22555  clsdif  22557  cmclsopn  22566  bcth3  24848
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