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Theorem ofcfn 33397
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 7445 . . 3 ((𝐹𝑥)𝑅𝐶) ∈ V
2 eqid 2731 . . 3 (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶))
31, 2fnmpti 6693 . 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 2732 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
84, 5, 6, 7ofcfval 33395 . . 3 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
98fneq1d 6642 . 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 2105  cmpt 5231   Fn wfn 6538  cfv 6543  (class class class)co 7412  f/c cofc 33392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ofc 33393
This theorem is referenced by:  probfinmeasb  33726  coinflipspace  33778
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