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Mirrors > Home > MPE Home > Th. List > Mathboxes > measbasedom | Structured version Visualization version Unicode version |
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measbasedom | measures measures |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrnmeas 30263 | . . . 4 measures sigAlgebra Disj Σ* | |
2 | 1 | simprd 479 | . . 3 measures Disj Σ* |
3 | dmmeas 30264 | . . . 4 measures sigAlgebra | |
4 | ismeas 30262 | . . . 4 sigAlgebra measures Disj Σ* | |
5 | 3, 4 | syl 17 | . . 3 measures measures Disj Σ* |
6 | 2, 5 | mpbird 247 | . 2 measures measures |
7 | df-meas 30259 | . . . 4 measures sigAlgebra Disj Σ* | |
8 | 7 | funmpt2 5927 | . . 3 measures |
9 | elunirn2 29451 | . . 3 measures measures measures | |
10 | 8, 9 | mpan 706 | . 2 measures measures |
11 | 6, 10 | impbii 199 | 1 measures measures |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cab 2608 wral 2912 c0 3915 cpw 4158 cuni 4436 Disj wdisj 4620 class class class wbr 4653 cdm 5114 crn 5115 wfun 5882 wf 5884 cfv 5888 (class class class)co 6650 com 7065 cdom 7953 cc0 9936 cpnf 10071 cicc 12178 Σ*cesum 30089 sigAlgebracsiga 30170 measurescmeas 30258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-esum 30090 df-meas 30259 |
This theorem is referenced by: truae 30306 aean 30307 mbfmbfm 30320 sibfinima 30401 sibfof 30402 domprobmeas 30472 probmeasd 30485 probfinmeasbOLD 30490 probfinmeasb 30491 probmeasb 30492 dstrvprob 30533 |
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