Theorem ofcfn 34611

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 7447	. . 3 ⊢ ((𝐹‘𝑥)𝑅𝐶) ∈ V
2		eqid 2760	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))
3	1, 2	fnmpti 6676	. 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 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
8	4, 5, 6, 7	ofcfval 34609	. . 3 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9	8	fneq1d 6626	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ↦ cmpt 5186   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414   ∘_f/c cofc 34606

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-ofc 34607

This theorem is used by:  probfinmeasb  34940  coinflipspace  34993