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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 22936 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 = ∪ 𝐽 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 ‘cfv 6563 TopOnctopon 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-topon 22933 |
This theorem is referenced by: toponrestid 22943 indisuni 23026 indistpsx 23033 letopuni 23231 dfac14 23642 unicntop 24822 sszcld 24853 reperflem 24854 cnperf 24856 iiuni 24921 abscncfALT 24965 cncfcnvcn 24966 cnheiborlem 25000 cnheibor 25001 cnllycmp 25002 bndth 25004 mbfimaopnlem 25704 limcnlp 25928 limcflflem 25930 limcflf 25931 limcmo 25932 limcres 25936 limccnp 25941 limccnp2 25942 perfdvf 25953 recnperf 25955 dvcnp2 25970 dvcnp2OLD 25971 dvaddbr 25989 dvmulbr 25990 dvmulbrOLD 25991 dvcobr 25998 dvcobrOLD 25999 dvcnvlem 26029 lhop1lem 26067 taylthlem2 26431 taylthlem2OLD 26432 abelth 26500 cxpcn3 26806 lgamucov 27096 ftalem3 27133 blocni 30834 ipasslem8 30866 ubthlem1 30899 tpr2uni 33866 tpr2rico 33873 mndpluscn 33887 raddcn 33890 cvxsconn 35228 cvmlift2lem11 35298 ivthALT 36318 poimir 37640 broucube 37641 dvtanlem 37656 ftc1cnnc 37679 dvasin 37691 dvacos 37692 dvreasin 37693 dvreacos 37694 areacirclem2 37696 reheibor 37826 islptre 45575 dirkercncf 46063 fourierdlem62 46124 |
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