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Theorem orvcval 31722
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 31721 . . 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 486 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
43cnveqd 5719 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑥 = 𝑋)
5 simpr 488 . . . . . 6 ((𝑥 = 𝑋𝑎 = 𝐴) → 𝑎 = 𝐴)
65breq2d 5051 . . . . 5 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑦𝑅𝑎𝑦𝑅𝐴))
76abbidv 2885 . . . 4 ((𝑥 = 𝑋𝑎 = 𝐴) → {𝑦𝑦𝑅𝑎} = {𝑦𝑦𝑅𝐴})
84, 7imaeq12d 5903 . . 3 ((𝑥 = 𝑋𝑎 = 𝐴) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
98adantl 485 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑎 = 𝐴)) → (𝑥 “ {𝑦𝑦𝑅𝑎}) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
10 orvcval.2 . . 3 (𝜑𝑋𝑉)
11 orvcval.1 . . 3 (𝜑 → Fun 𝑋)
12 funeq 6348 . . 3 (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋))
1310, 11, 12elabd 3646 . 2 (𝜑𝑋 ∈ {𝑥 ∣ Fun 𝑥})
14 orvcval.3 . . 3 (𝜑𝐴𝑊)
15 elex 3489 . . 3 (𝐴𝑊𝐴 ∈ V)
1614, 15syl 17 . 2 (𝜑𝐴 ∈ V)
17 cnvexg 7604 . . 3 (𝑋𝑉𝑋 ∈ V)
18 imaexg 7595 . . 3 (𝑋 ∈ V → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
1910, 17, 183syl 18 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) ∈ V)
202, 9, 13, 16, 19ovmpod 7276 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2799  Vcvv 3471   class class class wbr 5039  ccnv 5527  cima 5531  Fun wfun 6322  (class class class)co 7130  cmpo 7132  RV/𝑐corvc 31720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-orvc 31721
This theorem is referenced by:  orvcval2  31723  orvcval4  31725
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