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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 22427 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β ((clsβπ½)β(π β π΄)) = (π β ((intβπ½)βπ΄))) |
3 | 2 | ineq2d 4176 | . . 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 4234 | . . 3 β’ (((clsβπ½)βπ΄) β© (π β ((intβπ½)βπ΄))) = ((((clsβπ½)βπ΄) β© π) β ((intβπ½)βπ΄)) | |
5 | 3, 4 | eqtrdi 2789 | . 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 22433 | . . . 4 β’ ((π½ β Top β§ π΄ β π) β ((clsβπ½)βπ΄) β π) |
7 | df-ss 3931 | . . . 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 4085 | . 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 2773 | 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 397 = wceq 1542 β wcel 2107 β cdif 3911 β© cin 3913 β wss 3914 βͺ cuni 4869 βcfv 6500 Topctop 22265 intcnt 22391 clsccl 22392 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-top 22266 df-cld 22393 df-ntr 22394 df-cls 22395 |
This theorem is referenced by: opnbnd 34850 |
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