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Theorem orvcval 31614
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 31613 . . 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 483 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
43cnveqd 5739 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
5 simpr 485 . . . . . 6 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑎 = 𝐴)
65breq2d 5069 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑦𝑅𝑎𝑦𝑅𝐴))
76abbidv 2882 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → {𝑦𝑦𝑅𝑎} = {𝑦𝑦𝑅𝐴})
84, 7imaeq12d 5923 . . 3 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
98adantl 482 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑎 = 𝐴)) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
10 orvcval.2 . . 3 (𝜑𝑋𝑉)
11 orvcval.1 . . 3 (𝜑 → Fun 𝑋)
12 funeq 6368 . . 3 (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
1310, 11, 12elabd 3666 . 2 (𝜑𝑋 ∈ {𝑥 ∣ Fun 𝑥})
14 orvcval.3 . . 3 (𝜑𝐴𝑊)
15 elex 3510 . . 3 (𝐴𝑊𝐴 ∈ V)
1614, 15syl 17 . 2 (𝜑𝐴 ∈ V)
17 cnvexg 7618 . . 3 (𝑋𝑉𝑋 ∈ V)
18 imaexg 7609 . . 3 (𝑋 ∈ V → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
1910, 17, 183syl 18 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
202, 9, 13, 16, 19ovmpod 7291 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492   class class class wbr 5057  ccnv 5547  cima 5551  Fun wfun 6342  (class class class)co 7145  cmpo 7147  RV/𝑐corvc 31612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-orvc 31613
This theorem is referenced by:  orvcval2  31615  orvcval4  31617
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