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Mirrors > Home > MPE Home > Th. List > ustbas | Structured version Visualization version GIF version |
Description: Recover the base of an uniform structure π. βͺ ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
Ref | Expression |
---|---|
ustbas.1 | β’ π = dom βͺ π |
Ref | Expression |
---|---|
ustbas | β’ (π β βͺ ran UnifOn β π β (UnifOnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustfn 24057 | . . . 4 β’ UnifOn Fn V | |
2 | fnfun 6642 | . . . 4 β’ (UnifOn Fn V β Fun UnifOn) | |
3 | elunirn 7245 | . . . 4 β’ (Fun UnifOn β (π β βͺ ran UnifOn β βπ₯ β dom UnifOnπ β (UnifOnβπ₯))) | |
4 | 1, 2, 3 | mp2b 10 | . . 3 β’ (π β βͺ ran UnifOn β βπ₯ β dom UnifOnπ β (UnifOnβπ₯)) |
5 | ustbas2 24081 | . . . . . . . 8 β’ (π β (UnifOnβπ₯) β π₯ = dom βͺ π) | |
6 | ustbas.1 | . . . . . . . 8 β’ π = dom βͺ π | |
7 | 5, 6 | eqtr4di 2784 | . . . . . . 7 β’ (π β (UnifOnβπ₯) β π₯ = π) |
8 | 7 | fveq2d 6888 | . . . . . 6 β’ (π β (UnifOnβπ₯) β (UnifOnβπ₯) = (UnifOnβπ)) |
9 | 8 | eleq2d 2813 | . . . . 5 β’ (π β (UnifOnβπ₯) β (π β (UnifOnβπ₯) β π β (UnifOnβπ))) |
10 | 9 | ibi 267 | . . . 4 β’ (π β (UnifOnβπ₯) β π β (UnifOnβπ)) |
11 | 10 | rexlimivw 3145 | . . 3 β’ (βπ₯ β dom UnifOnπ β (UnifOnβπ₯) β π β (UnifOnβπ)) |
12 | 4, 11 | sylbi 216 | . 2 β’ (π β βͺ ran UnifOn β π β (UnifOnβπ)) |
13 | elfvunirn 6916 | . 2 β’ (π β (UnifOnβπ) β π β βͺ ran UnifOn) | |
14 | 12, 13 | impbii 208 | 1 β’ (π β βͺ ran UnifOn β π β (UnifOnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1533 β wcel 2098 βwrex 3064 Vcvv 3468 βͺ cuni 4902 dom cdm 5669 ran crn 5670 Fun wfun 6530 Fn wfn 6531 βcfv 6536 UnifOncust 24055 |
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-rn 5680 df-res 5681 df-iota 6488 df-fun 6538 df-fn 6539 df-fv 6544 df-ust 24056 |
This theorem is referenced by: (None) |
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