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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 6880 | . . 3 β’ (π΄ β (topGenβπ΅) β π΅ β dom topGen) | |
2 | eltg2b 22325 | . . . 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 3172 | . . . . 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 3577 | . . . 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 3061 βwrex 3070 β wss 3911 dom cdm 5634 βcfv 6497 topGenctg 17324 |
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 5257 ax-nul 5264 ax-pow 5321 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-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 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-iota 6449 df-fun 6499 df-fv 6505 df-topgen 17330 |
This theorem is referenced by: tgclb 22336 elcls3 22450 pnfnei 22587 mnfnei 22588 tgcnp 22620 tgcmp 22768 2ndcctbss 22822 2ndcdisj 22823 2ndcomap 22825 dis2ndc 22827 ptpjopn 22979 txlm 23015 flftg 23363 alexsublem 23411 alexsubALT 23418 tmdgsum2 23463 xrge0tsms 24213 xrge0tsmsd 31948 iccllysconn 33901 rellysconn 33902 fnessex 34864 ptrecube 36124 islptre 43946 |
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