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| Mirrors > Home > MPE Home > Th. List > ustimasn | Structured version Visualization version GIF version | ||
| Description: Lemma for ustuqtop 23650. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
| Ref | Expression |
|---|---|
| ustimasn | β’ ((π β (UnifOnβπ) β§ π β π β§ π β π) β (π β {π}) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imassrn 6044 | . 2 β’ (π β {π}) β ran π | |
| 2 | ustssxp 23608 | . . . 4 β’ ((π β (UnifOnβπ) β§ π β π) β π β (π Γ π)) | |
| 3 | 2 | 3adant3 1132 | . . 3 β’ ((π β (UnifOnβπ) β§ π β π β§ π β π) β π β (π Γ π)) |
| 4 | rnss 5914 | . . . 4 β’ (π β (π Γ π) β ran π β ran (π Γ π)) | |
| 5 | rnxpid 6145 | . . . 4 β’ ran (π Γ π) = π | |
| 6 | 4, 5 | sseqtrdi 4012 | . . 3 β’ (π β (π Γ π) β ran π β π) |
| 7 | 3, 6 | syl 17 | . 2 β’ ((π β (UnifOnβπ) β§ π β π β§ π β π) β ran π β π) |
| 8 | 1, 7 | sstrid 3973 | 1 β’ ((π β (UnifOnβπ) β§ π β π β§ π β π) β (π β {π}) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 β wcel 2106 β wss 3928 {csn 4606 Γ cxp 5651 ran crn 5654 β cima 5656 βcfv 6516 UnifOncust 23603 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fv 6524 df-ust 23604 |
| This theorem is referenced by: ustuqtop0 23644 ustuqtop4 23648 utopreg 23656 ucncn 23689 |
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