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Mirrors > Home > MPE Home > Th. List > basgen2 | Structured version Visualization version 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. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
basgen2 | β’ ((π½ β Top β§ π΅ β π½ β§ βπ₯ β π½ βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯)) β (topGenβπ΅) = π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3933 | . . . 4 β’ (π½ β (topGenβπ΅) β βπ₯ β π½ π₯ β (topGenβπ΅)) | |
2 | ssexg 5281 | . . . . . . 7 β’ ((π΅ β π½ β§ π½ β Top) β π΅ β V) | |
3 | 2 | ancoms 460 | . . . . . 6 β’ ((π½ β Top β§ π΅ β π½) β π΅ β V) |
4 | eltg2b 22325 | . . . . . 6 β’ (π΅ β V β (π₯ β (topGenβπ΅) β βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯))) | |
5 | 3, 4 | syl 17 | . . . . 5 β’ ((π½ β Top β§ π΅ β π½) β (π₯ β (topGenβπ΅) β βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯))) |
6 | 5 | ralbidv 3171 | . . . 4 β’ ((π½ β Top β§ π΅ β π½) β (βπ₯ β π½ π₯ β (topGenβπ΅) β βπ₯ β π½ βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯))) |
7 | 1, 6 | bitrid 283 | . . 3 β’ ((π½ β Top β§ π΅ β π½) β (π½ β (topGenβπ΅) β βπ₯ β π½ βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯))) |
8 | 7 | biimp3ar 1471 | . 2 β’ ((π½ β Top β§ π΅ β π½ β§ βπ₯ β π½ βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯)) β π½ β (topGenβπ΅)) |
9 | basgen 22354 | . 2 β’ ((π½ β Top β§ π΅ β π½ β§ π½ β (topGenβπ΅)) β (topGenβπ΅) = π½) | |
10 | 8, 9 | syld3an3 1410 | 1 β’ ((π½ β Top β§ π΅ β π½ β§ βπ₯ β π½ βπ¦ β π₯ βπ§ β π΅ (π¦ β π§ β§ π§ β π₯)) β (topGenβπ΅) = π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 Vcvv 3444 β wss 3911 βcfv 6497 topGenctg 17324 Topctop 22258 |
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 df-top 22259 |
This theorem is referenced by: pptbas 22374 2ndcctbss 22822 2ndcomap 22825 dis2ndc 22827 met2ndci 23894 |
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