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Mirrors > Home > MPE Home > Th. List > tg2 | Structured version Visualization version GIF version |
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
Ref | Expression |
---|---|
tg2 | β’ ((π΄ β (topGenβπ΅) β§ πΆ β π΄) β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6929 | . . 3 β’ (π΄ β (topGenβπ΅) β π΅ β dom topGen) | |
2 | eltg2b 22462 | . . . 4 β’ (π΅ β dom topGen β (π΄ β (topGenβπ΅) β βπ¦ β π΄ βπ₯ β π΅ (π¦ β π₯ β§ π₯ β π΄))) | |
3 | eleq1 2822 | . . . . . . 7 β’ (π¦ = πΆ β (π¦ β π₯ β πΆ β π₯)) | |
4 | 3 | anbi1d 631 | . . . . . 6 β’ (π¦ = πΆ β ((π¦ β π₯ β§ π₯ β π΄) β (πΆ β π₯ β§ π₯ β π΄))) |
5 | 4 | rexbidv 3179 | . . . . 5 β’ (π¦ = πΆ β (βπ₯ β π΅ (π¦ β π₯ β§ π₯ β π΄) β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄))) |
6 | 5 | rspccv 3610 | . . . 4 β’ (βπ¦ β π΄ βπ₯ β π΅ (π¦ β π₯ β§ π₯ β π΄) β (πΆ β π΄ β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄))) |
7 | 2, 6 | syl6bi 253 | . . 3 β’ (π΅ β dom topGen β (π΄ β (topGenβπ΅) β (πΆ β π΄ β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄)))) |
8 | 1, 7 | mpcom 38 | . 2 β’ (π΄ β (topGenβπ΅) β (πΆ β π΄ β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄))) |
9 | 8 | imp 408 | 1 β’ ((π΄ β (topGenβπ΅) β§ πΆ β π΄) β βπ₯ β π΅ (πΆ β π₯ β§ π₯ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 βwrex 3071 β wss 3949 dom cdm 5677 βcfv 6544 topGenctg 17383 |
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-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-topgen 17389 |
This theorem is referenced by: tgclb 22473 elcls3 22587 pnfnei 22724 mnfnei 22725 tgcnp 22757 tgcmp 22905 2ndcctbss 22959 2ndcdisj 22960 2ndcomap 22962 dis2ndc 22964 ptpjopn 23116 txlm 23152 flftg 23500 alexsublem 23548 alexsubALT 23555 tmdgsum2 23600 xrge0tsms 24350 xrge0tsmsd 32209 iccllysconn 34241 rellysconn 34242 fnessex 35231 ptrecube 36488 islptre 44335 |
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