Theorem dstrvval 30532

Description: The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)

Hypotheses

Ref	Expression
dstrvprob.1	|- ( ph -> P e. Prob )
dstrvprob.2	|- ( ph -> X e. (rRndVar ` P ) )
dstrvprob.3	|- ( ph -> D = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )
dstrvval.1	|- ( ph -> A e. 𝔅ℝ )

Assertion

Ref	Expression
dstrvval	|- ( ph -> ( D ` A ) = ( P ` ( `' X " A ) ) )

Distinct variable groups:   P, a   X, a   A, a

Allowed substitution hints:   ph( a)   D( a)

Proof of Theorem dstrvval

Step	Hyp	Ref	Expression
1		dstrvprob.3	. . 3 |- ( ph -> D = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) )
2	1	fveq1d 6193	. 2 |- ( ph -> ( D ` A ) = ( ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) ` A ) )
3		dstrvval.1	. . 3 |- ( ph -> A e. 𝔅ℝ )
4		oveq2 6658	. . . . 5 |- ( a = A -> ( X∘RV/𝑐 _E a ) = ( X∘RV/𝑐 _E A ) )
5	4	fveq2d 6195	. . . 4 |- ( a = A -> ( P ` ( X∘RV/𝑐 _E a ) ) = ( P ` ( X∘RV/𝑐 _E A ) ) )
6		eqid 2622	. . . 4 |- ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) = ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) )
7		fvex 6201	. . . 4 |- ( P ` ( X∘RV/𝑐 _E A ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . 3 |- ( A e. 𝔅ℝ -> ( ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) ` A ) = ( P ` ( X∘RV/𝑐 _E A ) ) )
9	3, 8	syl 17	. 2 |- ( ph -> ( ( a e. 𝔅ℝ |-> ( P ` ( X∘RV/𝑐 _E a ) ) ) ` A ) = ( P ` ( X∘RV/𝑐 _E A ) ) )
10		dstrvprob.1	. . . 4 |- ( ph -> P e. Prob )
11		dstrvprob.2	. . . 4 |- ( ph -> X e. (rRndVar ` P ) )
12	10, 11, 3	orvcelval 30530	. . 3 |- ( ph -> ( X∘RV/𝑐 _E A ) = ( `' X " A ) )
13	12	fveq2d 6195	. 2 |- ( ph -> ( P ` ( X∘RV/𝑐 _E A ) ) = ( P ` ( `' X " A ) ) )
14	2, 9, 13	3eqtrd 2660	1 |- ( ph -> ( D ` A ) = ( P ` ( `' X " A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   _E cep 5028   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650  𝔅_ℝcbrsiga 30244  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-eprel 5029  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:  dstrvprob  30533