Theorem ofcfn 34294

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 7392	. . 3 ⊢ ((𝐹‘𝑥)𝑅𝐶) ∈ V
2		eqid 2741	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))
3	1, 2	fnmpti 6631	. 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 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
8	4, 5, 6, 7	ofcfval 34292	. . 3 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9	8	fneq1d 6581	. 2 ⊢ (𝜑 → ((𝐹 ∘f/c 𝑅𝐶) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) Fn 𝐴))
10	3, 9	mpbiri 260	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2121   ↦ cmpt 5155   Fn wfn 6483  ‘cfv 6488  (class class class)co 7359   ∘_f/c cofc 34289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-ofc 34290

This theorem is referenced by:  probfinmeasb  34622  coinflipspace  34675