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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smffmptf | Structured version Visualization version GIF version |
Description: A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
Ref | Expression |
---|---|
smffmptf.x | β’ β²π₯π |
smffmptf.a | β’ β²π₯π΄ |
smffmptf.s | β’ (π β π β SAlg) |
smffmptf.b | β’ ((π β§ π₯ β π΄) β π΅ β π) |
smffmptf.m | β’ (π β (π₯ β π΄ β¦ π΅) β (SMblFnβπ)) |
Ref | Expression |
---|---|
smffmptf | β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smffmptf.s | . . 3 β’ (π β π β SAlg) | |
2 | smffmptf.m | . . 3 β’ (π β (π₯ β π΄ β¦ π΅) β (SMblFnβπ)) | |
3 | eqid 2728 | . . 3 β’ dom (π₯ β π΄ β¦ π΅) = dom (π₯ β π΄ β¦ π΅) | |
4 | 1, 2, 3 | smff 46167 | . 2 β’ (π β (π₯ β π΄ β¦ π΅):dom (π₯ β π΄ β¦ π΅)βΆβ) |
5 | smffmptf.x | . . . . 5 β’ β²π₯π | |
6 | smffmptf.a | . . . . 5 β’ β²π₯π΄ | |
7 | smffmptf.b | . . . . 5 β’ ((π β§ π₯ β π΄) β π΅ β π) | |
8 | 5, 6, 7 | dmmpt1 44692 | . . . 4 β’ (π β dom (π₯ β π΄ β¦ π΅) = π΄) |
9 | 8 | eqcomd 2734 | . . 3 β’ (π β π΄ = dom (π₯ β π΄ β¦ π΅)) |
10 | 9 | feq2d 6713 | . 2 β’ (π β ((π₯ β π΄ β¦ π΅):π΄βΆβ β (π₯ β π΄ β¦ π΅):dom (π₯ β π΄ β¦ π΅)βΆβ)) |
11 | 4, 10 | mpbird 256 | 1 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β²wnf 1777 β wcel 2098 β²wnfc 2879 β¦ cmpt 5235 dom cdm 5682 βΆwf 6549 βcfv 6553 βcr 11147 SAlgcsalg 45743 SMblFncsmblfn 46130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-pre-lttri 11222 ax-pre-lttrn 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-er 8733 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-ioo 13370 df-ico 13372 df-smblfn 46131 |
This theorem is referenced by: smffmpt 46240 smfdivdmmbl 46273 |
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