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Mirrors > Home > MPE Home > Th. List > Mathboxes > fneuni | Structured version Visualization version GIF version |
Description: If π΅ is finer than π΄, every element of π΄ is a union of elements of π΅. (Contributed by Jeff Hankins, 11-Oct-2009.) |
Ref | Expression |
---|---|
fneuni | β’ ((π΄Fneπ΅ β§ π β π΄) β βπ₯(π₯ β π΅ β§ π = βͺ π₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnetg 35738 | . . 3 β’ (π΄Fneπ΅ β π΄ β (topGenβπ΅)) | |
2 | 1 | sselda 3977 | . 2 β’ ((π΄Fneπ΅ β§ π β π΄) β π β (topGenβπ΅)) |
3 | elfvdm 6921 | . . . 4 β’ (π β (topGenβπ΅) β π΅ β dom topGen) | |
4 | eltg3 22816 | . . . 4 β’ (π΅ β dom topGen β (π β (topGenβπ΅) β βπ₯(π₯ β π΅ β§ π = βͺ π₯))) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π β (topGenβπ΅) β (π β (topGenβπ΅) β βπ₯(π₯ β π΅ β§ π = βͺ π₯))) |
6 | 5 | ibi 267 | . 2 β’ (π β (topGenβπ΅) β βπ₯(π₯ β π΅ β§ π = βͺ π₯)) |
7 | 2, 6 | syl 17 | 1 β’ ((π΄Fneπ΅ β§ π β π΄) β βπ₯(π₯ β π΅ β§ π = βͺ π₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 βwex 1773 β wcel 2098 β wss 3943 βͺ cuni 4902 class class class wbr 5141 dom cdm 5669 βcfv 6536 topGenctg 17390 Fnecfne 35729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-topgen 17396 df-fne 35730 |
This theorem is referenced by: (None) |
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