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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 20719 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 20718 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2622 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 20711 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2701 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 TopOnctopon 20715 |
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-top 20699 df-topon 20716 |
This theorem is referenced by: topgele 20734 eltpsg 20747 en2top 20789 resttopon 20965 ordtrest 21006 ordtrest2lem 21007 ordtrest2 21008 lmfval 21036 cnpfval 21038 iscn 21039 iscnp 21041 lmbrf 21064 cncls 21078 cnconst2 21087 cnrest2 21090 cndis 21095 cnindis 21096 cnpdis 21097 lmfss 21100 lmres 21104 lmff 21105 ist1-3 21153 connsuba 21223 unconn 21232 kgenval 21338 elkgen 21339 kgentopon 21341 pttoponconst 21400 tx1cn 21412 tx2cn 21413 ptcls 21419 xkoccn 21422 txlm 21451 cnmpt2res 21480 xkoinjcn 21490 qtoprest 21520 ordthmeolem 21604 pt1hmeo 21609 xkocnv 21617 flimclslem 21788 flfval 21794 flfnei 21795 isflf 21797 flfcnp 21808 txflf 21810 supnfcls 21824 fclscf 21829 fclscmp 21834 fcfval 21837 isfcf 21838 uffcfflf 21843 cnpfcf 21845 mopnm 22249 isxms2 22253 prdsxmslem2 22334 bcth2 23127 dvmptid 23720 dvmptc 23721 dvtaylp 24124 taylthlem1 24127 taylthlem2 24128 pige3 24269 dvcxp1 24481 cxpcn3 24489 ordtrestNEW 29967 ordtrest2NEWlem 29968 ordtrest2NEW 29969 topjoin 32360 areacirclem1 33500 |
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