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Theorem ofcfn 34101
Description: The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1 (𝜑𝐹 Fn 𝐴)
ofcfval.2 (𝜑𝐴𝑉)
ofcfval.3 (𝜑𝐶𝑊)
Assertion
Ref Expression
ofcfn (𝜑 → (𝐹f/c 𝑅𝐶) Fn 𝐴)

Proof of Theorem ofcfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 7464 . . 3 ((𝐹𝑥)𝑅𝐶) ∈ V
2 eqid 2737 . . 3 (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶))
31, 2fnmpti 6711 . 2 (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) Fn 𝐴
4 ofcfval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
5 ofcfval.2 . . . 4 (𝜑𝐴𝑉)
6 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
7 eqidd 2738 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
84, 5, 6, 7ofcfval 34099 . . 3 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
98fneq1d 6661 . 2 (𝜑 → ((𝐹f/c 𝑅𝐶) Fn 𝐴 ↔ (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) Fn 𝐴))
103, 9mpbiri 258 1 (𝜑 → (𝐹f/c 𝑅𝐶) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  cmpt 5225   Fn wfn 6556  cfv 6561  (class class class)co 7431  f/c cofc 34096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ofc 34097
This theorem is referenced by:  probfinmeasb  34430  coinflipspace  34483
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