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| 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 5927 | . 2 β’ ran πΌ = ran (intβπ½) |
| 3 | vpwex 5366 | . . . . . . . 8 β’ π« π β V | |
| 4 | 3 | inex2 5309 | . . . . . . 7 β’ (π½ β© π« π ) β V |
| 5 | 4 | uniex 7725 | . . . . . 6 β’ βͺ (π½ β© π« π ) β V |
| 6 | 5 | rgenw 3057 | . . . . 5 β’ βπ β π« πβͺ (π½ β© π« π ) β V |
| 7 | nfcv 2895 | . . . . . 6 β’ β²π π« π | |
| 8 | 7 | fnmptf 6677 | . . . . 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 22852 | . . . . 5 β’ (π½ β Top β (intβπ½) = (π β π« π β¦ βͺ (π½ β© π« π ))) |
| 12 | 11 | fneq1d 6633 | . . . 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 257 | . . 3 β’ (π½ β Top β (intβπ½) Fn π« π) |
| 14 | elpwi 4602 | . . . . 5 β’ (π β π« π β π β π) | |
| 15 | 10 | ntropn 22877 | . . . . . 6 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ ) β π½) |
| 16 | 15 | ex 412 | . . . . 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 3137 | . . 3 β’ (π½ β Top β βπ β π« π((intβπ½)βπ ) β π½) |
| 19 | fnfvrnss 7113 | . . 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 583 | . 2 β’ (π½ β Top β ran (intβπ½) β π½) |
| 21 | 2, 20 | eqsstrid 4023 | 1 β’ (π½ β Top β ran πΌ β π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3053 Vcvv 3466 β© cin 3940 β wss 3941 π« cpw 4595 βͺ cuni 4900 β¦ cmpt 5222 ran crn 5668 Fn wfn 6529 βcfv 6534 Topctop 22719 intcnt 22845 |
| 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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| 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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-top 22720 df-ntr 22848 |
| This theorem is referenced by: ntrf 43388 |
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