Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcfn Structured version   Visualization version   GIF version

Theorem ofcfn 33617
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 7435 . . 3 ((𝐹𝑥)𝑅𝐶) ∈ V
2 eqid 2724 . . 3 (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶))
31, 2fnmpti 6684 . 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 2725 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
84, 5, 6, 7ofcfval 33615 . . 3 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
98fneq1d 6633 . 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 2098  cmpt 5222   Fn wfn 6529  cfv 6534  (class class class)co 7402  f/c cofc 33612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-ofc 33613
This theorem is referenced by:  probfinmeasb  33946  coinflipspace  33998
  Copyright terms: Public domain W3C validator