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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 24119 | . . . 4 β’ UnifOn Fn V | |
2 | fnfun 6654 | . . . 4 β’ (UnifOn Fn V β Fun UnifOn) | |
3 | elunirn 7261 | . . . 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 24143 | . . . . . . . 8 β’ (π β (UnifOnβπ₯) β π₯ = dom βͺ π) | |
6 | ustbas.1 | . . . . . . . 8 β’ π = dom βͺ π | |
7 | 5, 6 | eqtr4di 2786 | . . . . . . 7 β’ (π β (UnifOnβπ₯) β π₯ = π) |
8 | 7 | fveq2d 6901 | . . . . . 6 β’ (π β (UnifOnβπ₯) β (UnifOnβπ₯) = (UnifOnβπ)) |
9 | 8 | eleq2d 2815 | . . . . 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 3148 | . . 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 6929 | . 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 1534 β wcel 2099 βwrex 3067 Vcvv 3471 βͺ cuni 4908 dom cdm 5678 ran crn 5679 Fun wfun 6542 Fn wfn 6543 βcfv 6548 UnifOncust 24117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-iota 6500 df-fun 6550 df-fn 6551 df-fv 6556 df-ust 24118 |
This theorem is referenced by: (None) |
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