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Mirrors > Home > MPE Home > Th. List > ustref | Structured version Visualization version GIF version |
Description: Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
Ref | Expression |
---|---|
ustref | β’ ((π β (UnifOnβπ) β§ π β π β§ π΄ β π) β π΄ππ΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 β’ π΄ = π΄ | |
2 | resieq 5949 | . . . . 5 β’ ((π΄ β π β§ π΄ β π) β (π΄( I βΎ π)π΄ β π΄ = π΄)) | |
3 | 1, 2 | mpbiri 258 | . . . 4 β’ ((π΄ β π β§ π΄ β π) β π΄( I βΎ π)π΄) |
4 | 3 | anidms 568 | . . 3 β’ (π΄ β π β π΄( I βΎ π)π΄) |
5 | 4 | 3ad2ant3 1136 | . 2 β’ ((π β (UnifOnβπ) β§ π β π β§ π΄ β π) β π΄( I βΎ π)π΄) |
6 | ustdiag 23576 | . . . 4 β’ ((π β (UnifOnβπ) β§ π β π) β ( I βΎ π) β π) | |
7 | 6 | ssbrd 5149 | . . 3 β’ ((π β (UnifOnβπ) β§ π β π) β (π΄( I βΎ π)π΄ β π΄ππ΄)) |
8 | 7 | 3adant3 1133 | . 2 β’ ((π β (UnifOnβπ) β§ π β π β§ π΄ β π) β (π΄( I βΎ π)π΄ β π΄ππ΄)) |
9 | 5, 8 | mpd 15 | 1 β’ ((π β (UnifOnβπ) β§ π β π β§ π΄ β π) β π΄ππ΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 UnifOncust 23567 |
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 2704 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-ust 23568 |
This theorem is referenced by: cstucnd 23652 |
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