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 5933 | . 2 β’ ran πΌ = ran (intβπ½) |
| 3 | vpwex 5371 | . . . . . . . 8 β’ π« π β V | |
| 4 | 3 | inex2 5312 | . . . . . . 7 β’ (π½ β© π« π ) β V |
| 5 | 4 | uniex 7740 | . . . . . 6 β’ βͺ (π½ β© π« π ) β V |
| 6 | 5 | rgenw 3061 | . . . . 5 β’ βπ β π« πβͺ (π½ β© π« π ) β V |
| 7 | nfcv 2899 | . . . . . 6 β’ β²π π« π | |
| 8 | 7 | fnmptf 6685 | . . . . 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 22921 | . . . . 5 β’ (π½ β Top β (intβπ½) = (π β π« π β¦ βͺ (π½ β© π« π ))) |
| 12 | 11 | fneq1d 6641 | . . . 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 4605 | . . . . 5 β’ (π β π« π β π β π) | |
| 15 | 10 | ntropn 22946 | . . . . . 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 3141 | . . 3 β’ (π½ β Top β βπ β π« π((intβπ½)βπ ) β π½) |
| 19 | fnfvrnss 7125 | . . 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 4026 | 1 β’ (π½ β Top β ran πΌ β π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βwral 3057 Vcvv 3470 β© cin 3944 β wss 3945 π« cpw 4598 βͺ cuni 4903 β¦ cmpt 5225 ran crn 5673 Fn wfn 6537 βcfv 6542 Topctop 22788 intcnt 22914 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22789 df-ntr 22917 |
| This theorem is referenced by: ntrf 43547 |
| Copyright terms: Public domain | W3C validator |