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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 6902 | . . . . . 6 β’ (π€ = π β (TopSetβπ€) = (TopSetβπ)) | |
2 | topnval.2 | . . . . . 6 β’ π½ = (TopSetβπ) | |
3 | 1, 2 | eqtr4di 2786 | . . . . 5 β’ (π€ = π β (TopSetβπ€) = π½) |
4 | fveq2 6902 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
5 | topnval.1 | . . . . . 6 β’ π΅ = (Baseβπ) | |
6 | 4, 5 | eqtr4di 2786 | . . . . 5 β’ (π€ = π β (Baseβπ€) = π΅) |
7 | 3, 6 | oveq12d 7444 | . . . 4 β’ (π€ = π β ((TopSetβπ€) βΎt (Baseβπ€)) = (π½ βΎt π΅)) |
8 | df-topn 17412 | . . . 4 β’ TopOpen = (π€ β V β¦ ((TopSetβπ€) βΎt (Baseβπ€))) | |
9 | ovex 7459 | . . . 4 β’ (π½ βΎt π΅) β V | |
10 | 7, 8, 9 | fvmpt 7010 | . . 3 β’ (π β V β (TopOpenβπ) = (π½ βΎt π΅)) |
11 | 10 | eqcomd 2734 | . 2 β’ (π β V β (π½ βΎt π΅) = (TopOpenβπ)) |
12 | 0rest 17418 | . . 3 β’ (β βΎt π΅) = β | |
13 | fvprc 6894 | . . . . 5 β’ (Β¬ π β V β (TopSetβπ) = β ) | |
14 | 2, 13 | eqtrid 2780 | . . . 4 β’ (Β¬ π β V β π½ = β ) |
15 | 14 | oveq1d 7441 | . . 3 β’ (Β¬ π β V β (π½ βΎt π΅) = (β βΎt π΅)) |
16 | fvprc 6894 | . . 3 β’ (Β¬ π β V β (TopOpenβπ) = β ) | |
17 | 12, 15, 16 | 3eqtr4a 2794 | . 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 3473 β c0 4326 βcfv 6553 (class class class)co 7426 Basecbs 17187 TopSetcts 17246 βΎt crest 17409 TopOpenctopn 17410 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-rest 17411 df-topn 17412 |
This theorem is referenced by: topnid 17424 topnpropd 17425 efmndtopn 18842 oppgtopn 19314 symgtopn 19368 mgptopn 20093 resstopn 23110 prdstopn 23552 tuslem 24191 tuslemOLD 24192 xrge0tsms 24770 om1opn 24983 xrge0tsmsd 32792 xrge0tmdALT 33580 |
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