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Theorem ofcfn 30678
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/𝑐𝑅𝐶) Fn 𝐴)

Proof of Theorem ofcfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 6910 . . 3 ((𝐹𝑥)𝑅𝐶) ∈ V
2 eqid 2799 . . 3 (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶))
31, 2fnmpti 6233 . 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 2800 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
84, 5, 6, 7ofcfval 30676 . . 3 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
98fneq1d 6192 . 2 (𝜑 → ((𝐹𝑓/𝑐𝑅𝐶) Fn 𝐴 ↔ (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) Fn 𝐴))
103, 9mpbiri 250 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  cmpt 4922   Fn wfn 6096  cfv 6101  (class class class)co 6878  𝑓/𝑐cofc 30673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-ofc 30674
This theorem is referenced by:  probfinmeasb  31008  coinflipspace  31059
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