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Mirrors > Home > MPE Home > Th. List > topnval | Structured version Visualization version GIF version |
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
topnval.1 | β’ π΅ = (Baseβπ) |
topnval.2 | β’ π½ = (TopSetβπ) |
Ref | Expression |
---|---|
topnval | β’ (π½ βΎt π΅) = (TopOpenβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6884 | . . . . . 6 β’ (π€ = π β (TopSetβπ€) = (TopSetβπ)) | |
2 | topnval.2 | . . . . . 6 β’ π½ = (TopSetβπ) | |
3 | 1, 2 | eqtr4di 2784 | . . . . 5 β’ (π€ = π β (TopSetβπ€) = π½) |
4 | fveq2 6884 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
5 | topnval.1 | . . . . . 6 β’ π΅ = (Baseβπ) | |
6 | 4, 5 | eqtr4di 2784 | . . . . 5 β’ (π€ = π β (Baseβπ€) = π΅) |
7 | 3, 6 | oveq12d 7422 | . . . 4 β’ (π€ = π β ((TopSetβπ€) βΎt (Baseβπ€)) = (π½ βΎt π΅)) |
8 | df-topn 17375 | . . . 4 β’ TopOpen = (π€ β V β¦ ((TopSetβπ€) βΎt (Baseβπ€))) | |
9 | ovex 7437 | . . . 4 β’ (π½ βΎt π΅) β V | |
10 | 7, 8, 9 | fvmpt 6991 | . . 3 β’ (π β V β (TopOpenβπ) = (π½ βΎt π΅)) |
11 | 10 | eqcomd 2732 | . 2 β’ (π β V β (π½ βΎt π΅) = (TopOpenβπ)) |
12 | 0rest 17381 | . . 3 β’ (β βΎt π΅) = β | |
13 | fvprc 6876 | . . . . 5 β’ (Β¬ π β V β (TopSetβπ) = β ) | |
14 | 2, 13 | eqtrid 2778 | . . . 4 β’ (Β¬ π β V β π½ = β ) |
15 | 14 | oveq1d 7419 | . . 3 β’ (Β¬ π β V β (π½ βΎt π΅) = (β βΎt π΅)) |
16 | fvprc 6876 | . . 3 β’ (Β¬ π β V β (TopOpenβπ) = β ) | |
17 | 12, 15, 16 | 3eqtr4a 2792 | . 2 β’ (Β¬ π β V β (π½ βΎt π΅) = (TopOpenβπ)) |
18 | 11, 17 | pm2.61i 182 | 1 β’ (π½ βΎt π΅) = (TopOpenβπ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3468 β c0 4317 βcfv 6536 (class class class)co 7404 Basecbs 17150 TopSetcts 17209 βΎt crest 17372 TopOpenctopn 17373 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-rest 17374 df-topn 17375 |
This theorem is referenced by: topnid 17387 topnpropd 17388 efmndtopn 18805 oppgtopn 19269 symgtopn 19323 mgptopn 20048 resstopn 23040 prdstopn 23482 tuslem 24121 tuslemOLD 24122 xrge0tsms 24700 om1opn 24913 xrge0tsmsd 32712 xrge0tmdALT 33455 |
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