![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sibff | Structured version Visualization version GIF version |
Description: A simple function is a function. (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 |
---|---|
sibff | β’ (π β πΉ:βͺ dom πβΆβͺ π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitgval.2 | . . . 4 β’ (π β π β βͺ ran measures) | |
2 | dmmeas 33199 | . . . 4 β’ (π β βͺ ran measures β dom π β βͺ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β dom π β βͺ ran sigAlgebra) |
4 | sitgval.s | . . . 4 β’ π = (sigaGenβπ½) | |
5 | sitgval.j | . . . . . 6 β’ π½ = (TopOpenβπ) | |
6 | fvexd 6907 | . . . . . 6 β’ (π β (TopOpenβπ) β V) | |
7 | 5, 6 | eqeltrid 2838 | . . . . 5 β’ (π β π½ β V) |
8 | 7 | sgsiga 33140 | . . . 4 β’ (π β (sigaGenβπ½) β βͺ ran sigAlgebra) |
9 | 4, 8 | eqeltrid 2838 | . . 3 β’ (π β π β βͺ ran sigAlgebra) |
10 | sitgval.b | . . . 4 β’ π΅ = (Baseβπ) | |
11 | sitgval.0 | . . . 4 β’ 0 = (0gβπ) | |
12 | sitgval.x | . . . 4 β’ Β· = ( Β·π βπ) | |
13 | sitgval.h | . . . 4 β’ π» = (βHomβ(Scalarβπ)) | |
14 | sitgval.1 | . . . 4 β’ (π β π β π) | |
15 | sibfmbl.1 | . . . 4 β’ (π β πΉ β dom (πsitgπ)) | |
16 | 10, 5, 4, 11, 12, 13, 14, 1, 15 | sibfmbl 33334 | . . 3 β’ (π β πΉ β (dom πMblFnMπ)) |
17 | 3, 9, 16 | mbfmf 33252 | . 2 β’ (π β πΉ:βͺ dom πβΆβͺ π) |
18 | 4 | unieqi 4922 | . . . 4 β’ βͺ π = βͺ (sigaGenβπ½) |
19 | unisg 33141 | . . . . 5 β’ (π½ β V β βͺ (sigaGenβπ½) = βͺ π½) | |
20 | 7, 19 | syl 17 | . . . 4 β’ (π β βͺ (sigaGenβπ½) = βͺ π½) |
21 | 18, 20 | eqtrid 2785 | . . 3 β’ (π β βͺ π = βͺ π½) |
22 | 21 | feq3d 6705 | . 2 β’ (π β (πΉ:βͺ dom πβΆβͺ π β πΉ:βͺ dom πβΆβͺ π½)) |
23 | 17, 22 | mpbid 231 | 1 β’ (π β πΉ:βͺ dom πβΆβͺ π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3475 βͺ cuni 4909 dom cdm 5677 ran crn 5678 βΆwf 6540 βcfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 TopOpenctopn 17367 0gc0g 17385 βHomcrrh 32973 sigAlgebracsiga 33106 sigaGencsigagen 33136 measurescmeas 33193 sitgcsitg 33328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-esum 33026 df-siga 33107 df-sigagen 33137 df-meas 33194 df-mbfm 33248 df-sitg 33329 |
This theorem is referenced by: sibfinima 33338 sibfof 33339 sitgaddlemb 33347 sitmcl 33350 |
Copyright terms: Public domain | W3C validator |