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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > topbnd | Structured version Visualization version GIF version |
Description: Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.) |
Ref | Expression |
---|---|
topbnd.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
topbnd | β’ ((π½ β Top β§ π΄ β π) β (((clsβπ½)βπ΄) β© ((clsβπ½)β(π β π΄))) = (((clsβπ½)βπ΄) β ((intβπ½)βπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topbnd.1 | . . . . 5 β’ π = βͺ π½ | |
2 | 1 | clsdif 22879 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β ((clsβπ½)β(π β π΄)) = (π β ((intβπ½)βπ΄))) |
3 | 2 | ineq2d 4204 | . . 3 β’ ((π½ β Top β§ π΄ β π) β (((clsβπ½)βπ΄) β© ((clsβπ½)β(π β π΄))) = (((clsβπ½)βπ΄) β© (π β ((intβπ½)βπ΄)))) |
4 | indif2 4262 | . . 3 β’ (((clsβπ½)βπ΄) β© (π β ((intβπ½)βπ΄))) = ((((clsβπ½)βπ΄) β© π) β ((intβπ½)βπ΄)) | |
5 | 3, 4 | eqtrdi 2780 | . 2 β’ ((π½ β Top β§ π΄ β π) β (((clsβπ½)βπ΄) β© ((clsβπ½)β(π β π΄))) = ((((clsβπ½)βπ΄) β© π) β ((intβπ½)βπ΄))) |
6 | 1 | clsss3 22885 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β ((clsβπ½)βπ΄) β π) |
7 | df-ss 3957 | . . . 4 β’ (((clsβπ½)βπ΄) β π β (((clsβπ½)βπ΄) β© π) = ((clsβπ½)βπ΄)) | |
8 | 6, 7 | sylib 217 | . . 3 β’ ((π½ β Top β§ π΄ β π) β (((clsβπ½)βπ΄) β© π) = ((clsβπ½)βπ΄)) |
9 | 8 | difeq1d 4113 | . 2 β’ ((π½ β Top β§ π΄ β π) β ((((clsβπ½)βπ΄) β© π) β ((intβπ½)βπ΄)) = (((clsβπ½)βπ΄) β ((intβπ½)βπ΄))) |
10 | 5, 9 | eqtrd 2764 | 1 β’ ((π½ β Top β§ π΄ β π) β (((clsβπ½)βπ΄) β© ((clsβπ½)β(π β π΄))) = (((clsβπ½)βπ΄) β ((intβπ½)βπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β cdif 3937 β© cin 3939 β wss 3940 βͺ cuni 4899 βcfv 6533 Topctop 22717 intcnt 22843 clsccl 22844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-top 22718 df-cld 22845 df-ntr 22846 df-cls 22847 |
This theorem is referenced by: opnbnd 35700 |
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