Nearby theorems
|
|
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrrn | Structured version Visualization version GIF version | ||
| Description: The range of the interior function of a topology a subset of the open sets of the topology. (Contributed by RP, 22-Apr-2021.) |
| Ref | Expression |
|---|---|
| ntrrn.x | β’ π = βͺ π½ |
| ntrrn.i | β’ πΌ = (intβπ½) |
| Ref | Expression |
|---|---|
| ntrrn | β’ (π½ β Top β ran πΌ β π½) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ntrrn.i | . . 3 β’ πΌ = (intβπ½) | |
| 2 | 1 | rneqi 5936 | . 2 β’ ran πΌ = ran (intβπ½) |
| 3 | vpwex 5375 | . . . . . . . 8 β’ π« π β V | |
| 4 | 3 | inex2 5318 | . . . . . . 7 β’ (π½ β© π« π ) β V |
| 5 | 4 | uniex 7730 | . . . . . 6 β’ βͺ (π½ β© π« π ) β V |
| 6 | 5 | rgenw 3065 | . . . . 5 β’ βπ β π« πβͺ (π½ β© π« π ) β V |
| 7 | nfcv 2903 | . . . . . 6 β’ β²π π« π | |
| 8 | 7 | fnmptf 6686 | . . . . 5 β’ (βπ β π« πβͺ (π½ β© π« π ) β V β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π) |
| 9 | 6, 8 | mp1i 13 | . . . 4 β’ (π½ β Top β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π) |
| 10 | ntrrn.x | . . . . . 6 β’ π = βͺ π½ | |
| 11 | 10 | ntrfval 22527 | . . . . 5 β’ (π½ β Top β (intβπ½) = (π β π« π β¦ βͺ (π½ β© π« π ))) |
| 12 | 11 | fneq1d 6642 | . . . 4 β’ (π½ β Top β ((intβπ½) Fn π« π β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π)) |
| 13 | 9, 12 | mpbird 256 | . . 3 β’ (π½ β Top β (intβπ½) Fn π« π) |
| 14 | elpwi 4609 | . . . . 5 β’ (π β π« π β π β π) | |
| 15 | 10 | ntropn 22552 | . . . . . 6 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ ) β π½) |
| 16 | 15 | ex 413 | . . . . 5 β’ (π½ β Top β (π β π β ((intβπ½)βπ ) β π½)) |
| 17 | 14, 16 | syl5 34 | . . . 4 β’ (π½ β Top β (π β π« π β ((intβπ½)βπ ) β π½)) |
| 18 | 17 | ralrimiv 3145 | . . 3 β’ (π½ β Top β βπ β π« π((intβπ½)βπ ) β π½) |
| 19 | fnfvrnss 7119 | . . 3 β’ (((intβπ½) Fn π« π β§ βπ β π« π((intβπ½)βπ ) β π½) β ran (intβπ½) β π½) | |
| 20 | 13, 18, 19 | syl2anc 584 | . 2 β’ (π½ β Top β ran (intβπ½) β π½) |
| 21 | 2, 20 | eqsstrid 4030 | 1 β’ (π½ β Top β ran πΌ β π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β© cin 3947 β wss 3948 π« cpw 4602 βͺ cuni 4908 β¦ cmpt 5231 ran crn 5677 Fn wfn 6538 βcfv 6543 Topctop 22394 intcnt 22520 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
| 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22395 df-ntr 22523 |
| This theorem is referenced by: ntrf 42864 |
| Copyright terms: Public domain | W3C validator |