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Mirrors > Home > MPE Home > Th. List > unitg | Structured version Visualization version Unicode version |
Description: The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.) |
Ref | Expression |
---|---|
unitg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tg1 20768 | . . . . . 6 | |
2 | selpw 4165 | . . . . . 6 | |
3 | 1, 2 | sylibr 224 | . . . . 5 |
4 | 3 | ssriv 3607 | . . . 4 |
5 | sspwuni 4611 | . . . 4 | |
6 | 4, 5 | mpbi 220 | . . 3 |
7 | 6 | a1i 11 | . 2 |
8 | bastg 20770 | . . 3 | |
9 | 8 | unissd 4462 | . 2 |
10 | 7, 9 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wss 3574 cpw 4158 cuni 4436 cfv 5888 ctg 16098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 |
This theorem is referenced by: tgcl 20773 tgtopon 20775 tgcmp 21204 2ndcsep 21262 txtopon 21394 ptuni 21397 xkouni 21402 prdstopn 21431 tgqtop 21515 alexsubb 21850 alexsubALTlem3 21853 alexsubALTlem4 21854 ptcmplem1 21856 uniretop 22566 fneval 32347 fnemeet1 32361 kelac2 37635 |
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