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| Mirrors > Home > ILE Home > Th. List > basgen | GIF version | ||
| Description: Given a topology π½, show that a subset π΅ satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
| Ref | Expression |
|---|---|
| basgen | β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ΅) = π½) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgss 13648 | . . . 4 β’ ((π½ β Top β§ π΅ β π½) β (topGenβπ΅) β (topGenβπ½)) | |
| 2 | 1 | 3adant3 1017 | . . 3 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ΅) β (topGenβπ½)) |
| 3 | tgtop 13653 | . . . 4 β’ (π½ β Top β (topGenβπ½) = π½) | |
| 4 | 3 | 3ad2ant1 1018 | . . 3 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ½) = π½) |
| 5 | 2, 4 | sseqtrd 3195 | . 2 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ΅) β π½) |
| 6 | simp3 999 | . 2 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β π½ β (topGenβπ΅)) | |
| 7 | 5, 6 | eqssd 3174 | 1 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ΅) = π½) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ w3a 978 = wceq 1353 β wcel 2148 β wss 3131 βcfv 5218 topGenctg 12708 Topctop 13582 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-topgen 12714 df-top 13583 |
| This theorem is referenced by: basgen2 13666 |
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