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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfrn | Structured version Visualization version GIF version |
Description: A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
sitgval.b | β’ π΅ = (Baseβπ) |
sitgval.j | β’ π½ = (TopOpenβπ) |
sitgval.s | β’ π = (sigaGenβπ½) |
sitgval.0 | β’ 0 = (0gβπ) |
sitgval.x | β’ Β· = ( Β·π βπ) |
sitgval.h | β’ π» = (βHomβ(Scalarβπ)) |
sitgval.1 | β’ (π β π β π) |
sitgval.2 | β’ (π β π β βͺ ran measures) |
sibfmbl.1 | β’ (π β πΉ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sibfrn | β’ (π β ran πΉ β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . 3 β’ (π β πΉ β dom (πsitgπ)) | |
2 | sitgval.b | . . . 4 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . 4 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . 4 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . 4 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . 4 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . 4 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . 4 β’ (π β π β π) | |
9 | sitgval.2 | . . . 4 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 33056 | . . 3 β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) |
11 | 1, 10 | mpbid 231 | . 2 β’ (π β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β))) |
12 | 11 | simp2d 1143 | 1 β’ (π β ran πΉ β Fin) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3060 β cdif 3925 {csn 4606 βͺ cuni 4885 β‘ccnv 5652 dom cdm 5653 ran crn 5654 β cima 5656 βcfv 6516 (class class class)co 7377 Fincfn 8905 0cc0 11075 +βcpnf 11210 [,)cico 13291 Basecbs 17109 Scalarcsca 17165 Β·π cvsca 17166 TopOpenctopn 17332 0gc0g 17350 βHomcrrh 32697 sigaGencsigagen 32860 measurescmeas 32917 MblFnMcmbfm 32971 sitgcsitg 33052 |
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-rep 5262 ax-sep 5276 ax-nul 5283 ax-pr 5404 |
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-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 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-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-sitg 33053 |
This theorem is referenced by: sibfof 33063 sitgfval 33064 sitgclg 33065 |
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