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| Mirrors > Home > MPE Home > Th. List > toponunii | Structured version Visualization version GIF version | ||
| Description: The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| topontopi.1 | ⊢ 𝐽 ∈ (TopOn‘𝐵) | 
| Ref | Expression | 
|---|---|
| toponunii | ⊢ 𝐵 = ∪ 𝐽 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | topontopi.1 | . 2 ⊢ 𝐽 ∈ (TopOn‘𝐵) | |
| 2 | toponuni 22921 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 = ∪ 𝐽 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 ‘cfv 6560 TopOnctopon 22917 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-topon 22918 | 
| This theorem is referenced by: toponrestid 22928 indisuni 23011 indistpsx 23018 letopuni 23216 dfac14 23627 unicntop 24807 sszcld 24840 reperflem 24841 cnperf 24843 iiuni 24908 abscncfALT 24952 cncfcnvcn 24953 cnheiborlem 24987 cnheibor 24988 cnllycmp 24989 bndth 24991 mbfimaopnlem 25691 limcnlp 25914 limcflflem 25916 limcflf 25917 limcmo 25918 limcres 25922 limccnp 25927 limccnp2 25928 perfdvf 25939 recnperf 25941 dvcnp2 25956 dvcnp2OLD 25957 dvaddbr 25975 dvmulbr 25976 dvmulbrOLD 25977 dvcobr 25984 dvcobrOLD 25985 dvcnvlem 26015 lhop1lem 26053 taylthlem2 26417 taylthlem2OLD 26418 abelth 26486 cxpcn3 26792 lgamucov 27082 ftalem3 27119 blocni 30825 ipasslem8 30857 ubthlem1 30890 tpr2uni 33905 tpr2rico 33912 mndpluscn 33926 raddcn 33929 cvxsconn 35249 cvmlift2lem11 35319 ivthALT 36337 poimir 37661 broucube 37662 ftc1cnnc 37700 dvasin 37712 dvacos 37713 dvreasin 37714 dvreacos 37715 areacirclem2 37717 reheibor 37847 islptre 45639 dirkercncf 46127 fourierdlem62 46188 | 
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