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Mirrors > Home > MPE Home > Th. List > toponmax | Structured version Visualization version GIF version |
Description: The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toponmax | ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 21665 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 21664 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2738 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 21657 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2833 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∪ cuni 4796 ‘cfv 6339 Topctop 21644 TopOnctopon 21661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-top 21645 df-topon 21662 |
This theorem is referenced by: topgele 21681 eltpsg 21694 en2top 21736 resttopon 21912 ordtrest 21953 ordtrest2lem 21954 ordtrest2 21955 lmfval 21983 cnpfval 21985 iscn 21986 iscnp 21988 lmbrf 22011 cncls 22025 cnconst2 22034 cnrest2 22037 cndis 22042 cnindis 22043 cnpdis 22044 lmfss 22047 lmres 22051 lmff 22052 ist1-3 22100 connsuba 22171 unconn 22180 kgenval 22286 elkgen 22287 kgentopon 22289 pttoponconst 22348 tx1cn 22360 tx2cn 22361 ptcls 22367 xkoccn 22370 txlm 22399 cnmpt2res 22428 xkoinjcn 22438 qtoprest 22468 ordthmeolem 22552 pt1hmeo 22557 xkocnv 22565 flimclslem 22735 flfval 22741 flfnei 22742 isflf 22744 flfcnp 22755 txflf 22757 supnfcls 22771 fclscf 22776 fclscmp 22781 fcfval 22784 isfcf 22785 uffcfflf 22790 cnpfcf 22792 mopnm 23197 isxms2 23201 prdsxmslem2 23282 bcth2 24082 dvmptid 24709 dvmptc 24710 dvtaylp 25117 taylthlem1 25120 taylthlem2 25121 pige3ALT 25264 dvcxp1 25481 cxpcn3 25489 ordtrestNEW 31443 ordtrest2NEWlem 31444 ordtrest2NEW 31445 topjoin 34192 areacirclem1 35488 |
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