Theorem ofcfn 34434

Description: The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofcfval.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfval.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
ofcfn	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) Fn 𝐴)

Proof of Theorem ofcfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7444	. . 3 ⊢ ((𝐹‘𝑥)𝑅𝐶) ∈ V
2		eqid 2769	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))
3	1, 2	fnmpti 6679	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) Fn 𝐴
4		ofcfval.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		ofcfval.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		ofcfval.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
7		eqidd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
8	4, 5, 6, 7	ofcfval 34432	. . 3 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9	8	fneq1d 6629	. 2 ⊢ (𝜑 → ((𝐹 ∘f/c 𝑅𝐶) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) Fn 𝐴))
10	3, 9	mpbiri 261	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149   ↦ cmpt 5196   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411   ∘_f/c cofc 34429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-ofc 34430

This theorem is referenced by:  probfinmeasb  34762  coinflipspace  34815