Theorem ofcfn 34090

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 7420	. . 3 ⊢ ((𝐹‘𝑥)𝑅𝐶) ∈ V
2		eqid 2729	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))
3	1, 2	fnmpti 6661	. 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 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
8	4, 5, 6, 7	ofcfval 34088	. . 3 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9	8	fneq1d 6611	. 2 ⊢ (𝜑 → ((𝐹 ∘f/c 𝑅𝐶) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) Fn 𝐴))
10	3, 9	mpbiri 258	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ↦ cmpt 5188   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∘_f/c cofc 34085

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-ofc 34086

This theorem is referenced by:  probfinmeasb  34419  coinflipspace  34472