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Mirrors > Home > MPE Home > Th. List > tgval3 | Structured version Visualization version GIF version |
Description: Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 22458 and tgval2 22459. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
tgval3 | β’ (π΅ β π β (topGenβπ΅) = {π₯ β£ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltg3 22465 | . 2 β’ (π΅ β π β (π₯ β (topGenβπ΅) β βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦))) | |
2 | 1 | eqabdv 2868 | 1 β’ (π΅ β π β (topGenβπ΅) = {π₯ β£ βπ¦(π¦ β π΅ β§ π₯ = βͺ π¦)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 {cab 2710 β wss 3949 βͺ cuni 4909 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: (None) |
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