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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 6843 | . . . . . 6 β’ (π€ = π β (TopSetβπ€) = (TopSetβπ)) | |
2 | topnval.2 | . . . . . 6 β’ π½ = (TopSetβπ) | |
3 | 1, 2 | eqtr4di 2791 | . . . . 5 β’ (π€ = π β (TopSetβπ€) = π½) |
4 | fveq2 6843 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
5 | topnval.1 | . . . . . 6 β’ π΅ = (Baseβπ) | |
6 | 4, 5 | eqtr4di 2791 | . . . . 5 β’ (π€ = π β (Baseβπ€) = π΅) |
7 | 3, 6 | oveq12d 7376 | . . . 4 β’ (π€ = π β ((TopSetβπ€) βΎt (Baseβπ€)) = (π½ βΎt π΅)) |
8 | df-topn 17310 | . . . 4 β’ TopOpen = (π€ β V β¦ ((TopSetβπ€) βΎt (Baseβπ€))) | |
9 | ovex 7391 | . . . 4 β’ (π½ βΎt π΅) β V | |
10 | 7, 8, 9 | fvmpt 6949 | . . 3 β’ (π β V β (TopOpenβπ) = (π½ βΎt π΅)) |
11 | 10 | eqcomd 2739 | . 2 β’ (π β V β (π½ βΎt π΅) = (TopOpenβπ)) |
12 | 0rest 17316 | . . 3 β’ (β βΎt π΅) = β | |
13 | fvprc 6835 | . . . . 5 β’ (Β¬ π β V β (TopSetβπ) = β ) | |
14 | 2, 13 | eqtrid 2785 | . . . 4 β’ (Β¬ π β V β π½ = β ) |
15 | 14 | oveq1d 7373 | . . 3 β’ (Β¬ π β V β (π½ βΎt π΅) = (β βΎt π΅)) |
16 | fvprc 6835 | . . 3 β’ (Β¬ π β V β (TopOpenβπ) = β ) | |
17 | 12, 15, 16 | 3eqtr4a 2799 | . 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 1542 β wcel 2107 Vcvv 3444 β c0 4283 βcfv 6497 (class class class)co 7358 Basecbs 17088 TopSetcts 17144 βΎt crest 17307 TopOpenctopn 17308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-rest 17309 df-topn 17310 |
This theorem is referenced by: topnid 17322 topnpropd 17323 efmndtopn 18698 oppgtopn 19139 symgtopn 19193 mgptopn 19913 resstopn 22553 prdstopn 22995 tuslem 23634 tuslemOLD 23635 xrge0tsms 24213 om1opn 24415 xrge0tsmsd 31948 xrge0tmdALT 32584 |
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