Theorem orvcval 30850

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 30849	. . 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 474	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑥 = 𝑋)
4	3	cnveqd 5454	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋)
5		simpr 479	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
6	5	breq2d 4817	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴))
7	6	abbidv 2880	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴})
8	4, 7	imaeq12d 5626	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
9	8	adantl 473	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑎 = 𝐴)) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
10		orvcval.1	. . 3 ⊢ (𝜑 → Fun 𝑋)
11		orvcval.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		funeq 6070	. . . . 5 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
13	12	elabg 3492	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋))
14	11, 13	syl 17	. . 3 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋))
15	10, 14	mpbird 247	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥})
16		orvcval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
17		elex 3353	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
19		cnvexg 7279	. . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V)
20		imaexg 7270	. . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V)
21	11, 19, 20	3syl 18	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V)
22	2, 9, 15, 18, 21	ovmpt2d 6955	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2140  {cab 2747  Vcvv 3341   class class class wbr 4805  ^◡ccnv 5266   “ cima 5270  Fun wfun 6044  (class class class)co 6815   ↦ cmpt2 6817  ∘_RV/𝑐corvc 30848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-orvc 30849

This theorem is referenced by:  orvcval2  30851  orvcval4  30853