![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ustrel | Structured version Visualization version GIF version |
Description: The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.) |
Ref | Expression |
---|---|
ustrel | β’ ((π β (UnifOnβπ) β§ π β π) β Rel π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustssxp 23572 | . . 3 β’ ((π β (UnifOnβπ) β§ π β π) β π β (π Γ π)) | |
2 | xpss 5650 | . . 3 β’ (π Γ π) β (V Γ V) | |
3 | 1, 2 | sstrdi 3957 | . 2 β’ ((π β (UnifOnβπ) β§ π β π) β π β (V Γ V)) |
4 | df-rel 5641 | . 2 β’ (Rel π β π β (V Γ V)) | |
5 | 3, 4 | sylibr 233 | 1 β’ ((π β (UnifOnβπ) β§ π β π) β Rel π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 Vcvv 3444 β wss 3911 Γ cxp 5632 Rel wrel 5639 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: ustssco 23582 ustexsym 23583 ustuqtop4 23612 utop2nei 23618 utop3cls 23619 ucncn 23653 |
Copyright terms: Public domain | W3C validator |