Theorem dstrvval

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

	Ref	Expression
Hypotheses	dstrvprob.1	$$⊢\phi \to P\in \mathrm{Prob}$$
	dstrvprob.2	$$⊢\phi \to X\in {\mathrm{RndVar}}_{ℝ}\left(P\right)$$
	dstrvprob.3	$$⊢\phi \to D=\left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)$$
	dstrvval.1	$$⊢\phi \to A\in {𝔅}_{ℝ}$$
Assertion	dstrvval	$$⊢\phi \to D\left(A\right)=P\left(X^{-1}\left[A\right]\right)$$

Proof

Step	Hyp	Ref	Expression
1		dstrvprob.1	$$⊢\phi \to P\in \mathrm{Prob}$$
2		dstrvprob.2	$$⊢\phi \to X\in {\mathrm{RndVar}}_{ℝ}\left(P\right)$$
3		dstrvprob.3	$$⊢\phi \to D=\left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)$$
4		dstrvval.1	$$⊢\phi \to A\in {𝔅}_{ℝ}$$
5	3	fveq1d	$$⊢\phi \to D\left(A\right)=\left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)\left(A\right)$$
6		oveq2	$$⊢a=A\to X{\mathrm{E}}_{\mathrm{RV/c}}a=X{\mathrm{E}}_{\mathrm{RV/c}}A$$
7	6	fveq2d	$$⊢a=A\to P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)=P\left(X{\mathrm{E}}_{\mathrm{RV/c}}A\right)$$
8		eqid	$$⊢\left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)=\left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)$$
9		fvex	$$⊢P\left(X{\mathrm{E}}_{\mathrm{RV/c}}A\right)\in \mathrm{V}$$
10	7 8 9	fvmpt	$$⊢A\in {𝔅}_{ℝ}\to \left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)\left(A\right)=P\left(X{\mathrm{E}}_{\mathrm{RV/c}}A\right)$$
11	4 10	syl	$$⊢\phi \to \left(a\in {𝔅}_{ℝ}⟼P\left(X{\mathrm{E}}_{\mathrm{RV/c}}a\right)\right)\left(A\right)=P\left(X{\mathrm{E}}_{\mathrm{RV/c}}A\right)$$
12	1 2 4	orvcelval	$$⊢\phi \to X{\mathrm{E}}_{\mathrm{RV/c}}A=X^{-1}\left[A\right]$$
13	12	fveq2d	$$⊢\phi \to P\left(X{\mathrm{E}}_{\mathrm{RV/c}}A\right)=P\left(X^{-1}\left[A\right]\right)$$
14	5 11 13	3eqtrd	$$⊢\phi \to D\left(A\right)=P\left(X^{-1}\left[A\right]\right)$$