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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem2 | Structured version Visualization version GIF version |
Description: Lemma for kur14 31198. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | ⊢ 𝐽 ∈ Top |
kur14lem.x | ⊢ 𝑋 = ∪ 𝐽 |
kur14lem.k | ⊢ 𝐾 = (cls‘𝐽) |
kur14lem.i | ⊢ 𝐼 = (int‘𝐽) |
kur14lem.a | ⊢ 𝐴 ⊆ 𝑋 |
Ref | Expression |
---|---|
kur14lem2 | ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.j | . . 3 ⊢ 𝐽 ∈ Top | |
2 | kur14lem.a | . . 3 ⊢ 𝐴 ⊆ 𝑋 | |
3 | kur14lem.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | ntrval2 20855 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) |
5 | 1, 2, 4 | mp2an 708 | . 2 ⊢ ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) |
6 | kur14lem.i | . . 3 ⊢ 𝐼 = (int‘𝐽) | |
7 | 6 | fveq1i 6192 | . 2 ⊢ (𝐼‘𝐴) = ((int‘𝐽)‘𝐴) |
8 | kur14lem.k | . . . 4 ⊢ 𝐾 = (cls‘𝐽) | |
9 | 8 | fveq1i 6192 | . . 3 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) |
10 | 9 | difeq2i 3725 | . 2 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) |
11 | 5, 7, 10 | 3eqtr4i 2654 | 1 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ⊆ wss 3574 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 intcnt 20821 clsccl 20822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-top 20699 df-cld 20823 df-ntr 20824 df-cls 20825 |
This theorem is referenced by: kur14lem6 31193 kur14lem7 31194 |
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