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Mirrors > Home > MPE Home > Th. List > ustbasel | Structured version Visualization version GIF version |
Description: The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
Ref | Expression |
---|---|
ustbasel | β’ (π β (UnifOnβπ) β (π Γ π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6881 | . . . 4 β’ (π β (UnifOnβπ) β π β V) | |
2 | isust 23571 | . . . 4 β’ (π β V β (π β (UnifOnβπ) β (π β π« (π Γ π) β§ (π Γ π) β π β§ βπ£ β π (βπ€ β π« (π Γ π)(π£ β π€ β π€ β π) β§ βπ€ β π (π£ β© π€) β π β§ (( I βΎ π) β π£ β§ β‘π£ β π β§ βπ€ β π (π€ β π€) β π£))))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β (UnifOnβπ) β (π β (UnifOnβπ) β (π β π« (π Γ π) β§ (π Γ π) β π β§ βπ£ β π (βπ€ β π« (π Γ π)(π£ β π€ β π€ β π) β§ βπ€ β π (π£ β© π€) β π β§ (( I βΎ π) β π£ β§ β‘π£ β π β§ βπ€ β π (π€ β π€) β π£))))) |
4 | 3 | ibi 267 | . 2 β’ (π β (UnifOnβπ) β (π β π« (π Γ π) β§ (π Γ π) β π β§ βπ£ β π (βπ€ β π« (π Γ π)(π£ β π€ β π€ β π) β§ βπ€ β π (π£ β© π€) β π β§ (( I βΎ π) β π£ β§ β‘π£ β π β§ βπ€ β π (π€ β π€) β π£)))) |
5 | 4 | simp2d 1144 | 1 β’ (π β (UnifOnβπ) β (π Γ π) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 β wcel 2107 βwral 3061 βwrex 3070 Vcvv 3444 β© cin 3910 β wss 3911 π« cpw 4561 I cid 5531 Γ cxp 5632 β‘ccnv 5633 βΎ cres 5636 β ccom 5638 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: ustne0 23581 ust0 23587 ustbas2 23593 ustuni 23594 trust 23597 ustuqtop5 23613 |
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