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Theorem orvcval 30300
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 30299 . . 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 473 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
43cnveqd 5258 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
5 simpr 477 . . . . . 6 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑎 = 𝐴)
65breq2d 4625 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑦𝑅𝑎𝑦𝑅𝐴))
76abbidv 2738 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → {𝑦𝑦𝑅𝑎} = {𝑦𝑦𝑅𝐴})
84, 7imaeq12d 5426 . . 3 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
98adantl 482 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑎 = 𝐴)) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
10 orvcval.1 . . 3 (𝜑 → Fun 𝑋)
11 orvcval.2 . . . 4 (𝜑𝑋𝑉)
12 funeq 5867 . . . . 5 (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
1312elabg 3334 . . . 4 (𝑋𝑉 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋))
1411, 13syl 17 . . 3 (𝜑 → (𝑋 ∈ {𝑥 ∣ Fun 𝑥} ↔ Fun 𝑋))
1510, 14mpbird 247 . 2 (𝜑𝑋 ∈ {𝑥 ∣ Fun 𝑥})
16 orvcval.3 . . 3 (𝜑𝐴𝑊)
17 elex 3198 . . 3 (𝐴𝑊𝐴 ∈ V)
1816, 17syl 17 . 2 (𝜑𝐴 ∈ V)
19 cnvexg 7059 . . 3 (𝑋𝑉𝑋 ∈ V)
20 imaexg 7050 . . 3 (𝑋 ∈ V → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
2111, 19, 203syl 18 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
222, 9, 15, 18, 21ovmpt2d 6741 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186   class class class wbr 4613  ccnv 5073  cima 5077  Fun wfun 5841  (class class class)co 6604  cmpt2 6606  RV/𝑐corvc 30298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-orvc 30299
This theorem is referenced by:  orvcval2  30301  orvcval4  30303
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