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Theorem orvcval 34439
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 (𝜑𝑋𝑉)
orvcval.3 (𝜑𝐴𝑊)
Assertion
Ref Expression
orvcval (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
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.
StepHypRef Expression
1 df-orvc 34438 . . 3 RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
21a1i 11 . 2 (𝜑 → ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎})))
3 simpl 482 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
43cnveqd 5889 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
5 simpr 484 . . . . . 6 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑎 = 𝐴)
65breq2d 5160 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑦𝑅𝑎𝑦𝑅𝐴))
76abbidv 2806 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → {𝑦𝑦𝑅𝑎} = {𝑦𝑦𝑅𝐴})
84, 7imaeq12d 6081 . . 3 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
98adantl 481 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑎 = 𝐴)) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
10 orvcval.2 . . 3 (𝜑𝑋𝑉)
11 orvcval.1 . . 3 (𝜑 → Fun 𝑋)
12 funeq 6588 . . 3 (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
1310, 11, 12elabd 3684 . 2 (𝜑𝑋 ∈ {𝑥 ∣ Fun 𝑥})
14 orvcval.3 . . 3 (𝜑𝐴𝑊)
15 elex 3499 . . 3 (𝐴𝑊𝐴 ∈ V)
1614, 15syl 17 . 2 (𝜑𝐴 ∈ V)
17 cnvexg 7947 . . 3 (𝑋𝑉𝑋 ∈ V)
18 imaexg 7936 . . 3 (𝑋 ∈ V → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
1910, 17, 183syl 18 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
202, 9, 13, 16, 19ovmpod 7585 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478   class class class wbr 5148  ccnv 5688  cima 5692  Fun wfun 6557  (class class class)co 7431  cmpo 7433  RV/𝑐corvc 34437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-orvc 34438
This theorem is referenced by:  orvcval2  34440  orvcval4  34442
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