dstrvval - Mathbox for Thierry Arnoux

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Theorem dstrvval 34508

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

**Hypotheses**
Ref	Expression
dstrvprob.1	⊢ (𝜑 → 𝑃 ∈ Prob)
dstrvprob.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
dstrvprob.3	⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
dstrvval.1	⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)

**Assertion**
Ref	Expression
dstrvval	⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Distinct variable groups: 𝑃,𝑎 𝑋,𝑎 𝐴,𝑎 Allowed substitution hints: 𝜑(𝑎) 𝐷(𝑎)

**Proof of Theorem dstrvval**
Step	Hyp	Ref	Expression
1		dstrvprob.3	. . 3 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))))
2	1	fveq1d 6883	. 2 ⊢ (𝜑 → (𝐷‘𝐴) = ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴))
3		dstrvval.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅_ℝ)
4		oveq2 7418	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑋∘_RV/𝑐 E 𝑎) = (𝑋∘_RV/𝑐 E 𝐴))
5	4	fveq2d 6885	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
6		eqid 2736	. . . 4 ⊢ (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎))) = (𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))
7		fvex 6894	. . . 4 ⊢ (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) ∈ V
8	5, 6, 7	fvmpt 6991	. . 3 ⊢ (𝐴 ∈ 𝔅_ℝ → ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
9	3, 8	syl 17	. 2 ⊢ (𝜑 → ((𝑎 ∈ 𝔅_ℝ ↦ (𝑃‘(𝑋∘_RV/𝑐 E 𝑎)))‘𝐴) = (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)))
10		dstrvprob.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob)
11		dstrvprob.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
12	10, 11, 3	orvcelval 34506	. . 3 ⊢ (𝜑 → (𝑋∘_RV/𝑐 E 𝐴) = (^◡𝑋 “ 𝐴))
13	12	fveq2d 6885	. 2 ⊢ (𝜑 → (𝑃‘(𝑋∘_RV/𝑐 E 𝐴)) = (𝑃‘(^◡𝑋 “ 𝐴)))
14	2, 9, 13	3eqtrd 2775	1 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(^◡𝑋 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ↦ cmpt 5206 E cep 5557 ^◡ccnv 5658 “ cima 5662 ‘cfv 6536 (class class class)co 7410 𝔅_ℝcbrsiga 34217 Probcprb 34444 rRndVarcrrv 34477 ∘_RV/𝑐corvc 34493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-pre-lttri 11208 ax-pre-lttrn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-ioo 13371 df-topgen 17462 df-top 22837 df-bases 22889 df-esum 34064 df-siga 34145 df-sigagen 34175 df-brsiga 34218 df-meas 34232 df-mbfm 34286 df-prob 34445 df-rrv 34478 df-orvc 34494

This theorem is referenced by: dstrvprob 34509

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