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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 21450 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐵 = ∪ 𝐽 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 ‘cfv 6348 TopOnctopon 21446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-topon 21447 |
This theorem is referenced by: toponrestid 21457 indisuni 21539 indistpsx 21546 letopuni 21743 dfac14 22154 unicntop 23321 sszcld 23352 reperflem 23353 cnperf 23355 iiuni 23416 abscncfALT 23455 cncfcnvcn 23456 cnheiborlem 23485 cnheibor 23486 cnllycmp 23487 bndth 23489 mbfimaopnlem 24183 limcnlp 24403 limcflflem 24405 limcflf 24406 limcmo 24407 limcres 24411 limccnp 24416 limccnp2 24417 perfdvf 24428 recnperf 24430 dvcnp2 24444 dvaddbr 24462 dvmulbr 24463 dvcobr 24470 dvcnvlem 24500 lhop1lem 24537 taylthlem2 24889 abelth 24956 cxpcn3 25256 lgamucov 25542 ftalem3 25579 blocni 28509 ipasslem8 28541 ubthlem1 28574 tpr2uni 31047 tpr2rico 31054 mndpluscn 31068 rmulccn 31070 raddcn 31071 cvxsconn 32387 cvmlift2lem11 32457 ivthALT 33580 poimir 34806 broucube 34807 dvtanlem 34822 ftc1cnnc 34847 dvasin 34859 dvacos 34860 dvreasin 34861 dvreacos 34862 areacirclem2 34864 reheibor 34998 islptre 41776 dirkercncf 42269 fourierdlem62 42330 |
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