Theorem orvcval 34490

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 34489	. . 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 5855	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋)
5		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
6	5	breq2d 5131	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴))
7	6	abbidv 2801	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴})
8	4, 7	imaeq12d 6048	. . 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 6556	. . 3 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
13	10, 11, 12	elabd 3660	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥})
14		orvcval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
15		elex 3480	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
17		cnvexg 7920	. . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V)
18		imaexg 7909	. . 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 7559	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  Vcvv 3459   class class class wbr 5119  ^◡ccnv 5653   “ cima 5657  Fun wfun 6525  (class class class)co 7405   ∈ cmpo 7407  ∘_RV/𝑐corvc 34488

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-orvc 34489

This theorem is referenced by:  orvcval2  34491  orvcval4  34493