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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 2726 | . . 3 β’ dom (π₯ β π΄ β¦ π΅) = dom (π₯ β π΄ β¦ π΅) | |
4 | 1, 2, 3 | smff 46017 | . 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 44542 | . . . 4 β’ (π β dom (π₯ β π΄ β¦ π΅) = π΄) |
9 | 8 | eqcomd 2732 | . . 3 β’ (π β π΄ = dom (π₯ β π΄ β¦ π΅)) |
10 | 9 | feq2d 6697 | . 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 257 | 1 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β²wnf 1777 β wcel 2098 β²wnfc 2877 β¦ cmpt 5224 dom cdm 5669 βΆwf 6533 βcfv 6537 βcr 11111 SAlgcsalg 45593 SMblFncsmblfn 45980 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13334 df-ico 13336 df-smblfn 45981 |
This theorem is referenced by: smffmpt 46090 smfdivdmmbl 46123 |
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