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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 23556 | . . . 4 β’ UnifOn Fn V | |
2 | fnfun 6603 | . . . 4 β’ (UnifOn Fn V β Fun UnifOn) | |
3 | elunirn 7199 | . . . 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 23580 | . . . . . . . 8 β’ (π β (UnifOnβπ₯) β π₯ = dom βͺ π) | |
6 | ustbas.1 | . . . . . . . 8 β’ π = dom βͺ π | |
7 | 5, 6 | eqtr4di 2795 | . . . . . . 7 β’ (π β (UnifOnβπ₯) β π₯ = π) |
8 | 7 | fveq2d 6847 | . . . . . 6 β’ (π β (UnifOnβπ₯) β (UnifOnβπ₯) = (UnifOnβπ)) |
9 | 8 | eleq2d 2824 | . . . . 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 3149 | . . 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 6875 | . 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 1542 β wcel 2107 βwrex 3074 Vcvv 3446 βͺ cuni 4866 dom cdm 5634 ran crn 5635 Fun wfun 6491 Fn wfn 6492 βcfv 6497 UnifOncust 23554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ust 23555 |
This theorem is referenced by: (None) |
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