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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 20941 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 = ∪ 𝐽 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ∪ cuni 4588 ‘cfv 6049 TopOnctopon 20937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-topon 20938 |
This theorem is referenced by: indisuni 21029 indistpsx 21036 letopuni 21233 dfac14 21643 unicntop 22810 sszcld 22841 reperflem 22842 cnperf 22844 iiuni 22905 cncfcn1 22934 cncfmpt2f 22938 cdivcncf 22941 abscncfALT 22944 cncfcnvcn 22945 cnrehmeo 22973 cnheiborlem 22974 cnheibor 22975 cnllycmp 22976 bndth 22978 csscld 23268 clsocv 23269 cncmet 23339 resscdrg 23374 mbfimaopnlem 23641 limcnlp 23861 limcflflem 23863 limcflf 23864 limcmo 23865 limcres 23869 cnlimc 23871 limccnp 23874 limccnp2 23875 limciun 23877 perfdvf 23886 recnperf 23888 dvidlem 23898 dvcnp2 23902 dvcn 23903 dvnres 23913 dvaddbr 23920 dvmulbr 23921 dvcobr 23928 dvcjbr 23931 dvrec 23937 dvcnvlem 23958 dvexp3 23960 dveflem 23961 dvlipcn 23976 lhop1lem 23995 ftc1cn 24025 dvply1 24258 dvtaylp 24343 taylthlem2 24347 psercn 24399 pserdvlem2 24401 pserdv 24402 abelth 24414 logcn 24613 dvloglem 24614 logdmopn 24615 dvlog 24617 dvlog2 24619 efopnlem2 24623 logtayl 24626 cxpcn 24706 cxpcn2 24707 cxpcn3 24709 resqrtcn 24710 sqrtcn 24711 dvatan 24882 efrlim 24916 lgamucov 24984 lgamucov2 24985 ftalem3 25021 blocni 27990 ipasslem8 28022 ubthlem1 28056 tpr2uni 30281 tpr2rico 30288 mndpluscn 30302 rmulccn 30304 raddcn 30305 cxpcncf1 31003 cvxsconn 31553 cvmlift2lem11 31623 ivthALT 32657 knoppcnlem10 32819 knoppcnlem11 32820 poimir 33773 broucube 33774 dvtanlem 33790 dvtan 33791 ftc1cnnc 33815 dvasin 33827 dvacos 33828 dvreasin 33829 dvreacos 33830 areacirclem2 33832 reheibor 33969 islptre 40372 cxpcncf2 40634 dirkercncf 40845 fourierdlem62 40906 |
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