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| Mirrors > Home > MPE Home > Th. List > ustimasn | Structured version Visualization version GIF version | ||
| Description: Lemma for ustuqtop 23972. (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 6070 | . 2 β’ (π β {π}) β ran π | |
| 2 | ustssxp 23930 | . . . 4 β’ ((π β (UnifOnβπ) β§ π β π) β π β (π Γ π)) | |
| 3 | 2 | 3adant3 1131 | . . 3 β’ ((π β (UnifOnβπ) β§ π β π β§ π β π) β π β (π Γ π)) |
| 4 | rnss 5938 | . . . 4 β’ (π β (π Γ π) β ran π β ran (π Γ π)) | |
| 5 | rnxpid 6172 | . . . 4 β’ ran (π Γ π) = π | |
| 6 | 4, 5 | sseqtrdi 4032 | . . 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 3993 | 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 1086 β wcel 2105 β wss 3948 {csn 4628 Γ cxp 5674 ran crn 5677 β cima 5679 βcfv 6543 UnifOncust 23925 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ust 23926 |
| This theorem is referenced by: ustuqtop0 23966 ustuqtop4 23970 utopreg 23978 ucncn 24011 |
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