Theorem orvcval 32403

Description: Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.)

Hypotheses

Ref	Expression
orvcval.1	⊢ (𝜑 → Fun 𝑋)
orvcval.2	⊢ (𝜑 → 𝑋 ∈ 𝑉)
orvcval.3	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
orvcval	⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem orvcval

Dummy variables 𝑥 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-orvc 32402	. . 3 ⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
2	1	a1i 11	. 2 ⊢ (𝜑 → ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})))
3		simpl 482	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑥 = 𝑋)
4	3	cnveqd 5781	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋)
5		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
6	5	breq2d 5090	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴))
7	6	abbidv 2808	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴})
8	4, 7	imaeq12d 5967	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
9	8	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑎 = 𝐴)) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
10		orvcval.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
11		orvcval.1	. . 3 ⊢ (𝜑 → Fun 𝑋)
12		funeq 6450	. . 3 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
13	10, 11, 12	elabd 3613	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥})
14		orvcval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
15		elex 3448	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
17		cnvexg 7758	. . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V)
18		imaexg 7749	. . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V)
19	10, 17, 18	3syl 18	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V)
20	2, 9, 13, 16, 19	ovmpod 7416	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  {cab 2716  Vcvv 3430   class class class wbr 5078  ^◡ccnv 5587   “ cima 5591  Fun wfun 6424  (class class class)co 7268   ∈ cmpo 7270  ∘_RV/𝑐corvc 32401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-orvc 32402

This theorem is referenced by:  orvcval2  32404  orvcval4  32406