Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orrvcval4 | Structured version Visualization version Unicode version |
Description: The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orrvccel.1 | Prob |
orrvccel.2 | rRndVar |
orrvccel.4 |
Ref | Expression |
---|---|
orrvcval4 | ∘RV/𝑐 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . . 4 Prob | |
2 | domprobsiga 30473 | . . . 4 Prob sigAlgebra | |
3 | 1, 2 | syl 17 | . . 3 sigAlgebra |
4 | retop 22565 | . . . 4 | |
5 | 4 | a1i 11 | . . 3 |
6 | orrvccel.2 | . . . . 5 rRndVar | |
7 | 1 | rrvmbfm 30504 | . . . . 5 rRndVar MblFnM𝔅ℝ |
8 | 6, 7 | mpbid 222 | . . . 4 MblFnM𝔅ℝ |
9 | df-brsiga 30245 | . . . . 5 𝔅ℝ sigaGen | |
10 | 9 | oveq2i 6661 | . . . 4 MblFnM𝔅ℝ MblFnMsigaGen |
11 | 8, 10 | syl6eleq 2711 | . . 3 MblFnMsigaGen |
12 | orrvccel.4 | . . 3 | |
13 | 3, 5, 11, 12 | orvcval4 30522 | . 2 ∘RV/𝑐 |
14 | uniretop 22566 | . . . 4 | |
15 | rabeq 3192 | . . . 4 | |
16 | 14, 15 | ax-mp 5 | . . 3 |
17 | 16 | imaeq2i 5464 | . 2 |
18 | 13, 17 | syl6eqr 2674 | 1 ∘RV/𝑐 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 crab 2916 cuni 4436 class class class wbr 4653 ccnv 5113 cdm 5114 crn 5115 cima 5117 cfv 5888 (class class class)co 6650 cr 9935 cioo 12175 ctg 16098 ctop 20698 sigAlgebracsiga 30170 sigaGencsigagen 30201 𝔅ℝcbrsiga 30244 MblFnMcmbfm 30312 Probcprb 30469 rRndVarcrrv 30502 ∘RV/𝑐corvc 30517 |
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 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 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-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-topgen 16104 df-top 20699 df-bases 20750 df-esum 30090 df-siga 30171 df-sigagen 30202 df-brsiga 30245 df-meas 30259 df-mbfm 30313 df-prob 30470 df-rrv 30503 df-orvc 30518 |
This theorem is referenced by: orvcelval 30530 dstfrvel 30535 orvclteinc 30537 |
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