Theorem ofcfn 29489

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/𝑐𝑅𝐶) Fn 𝐴)

Proof of Theorem ofcfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6577	. . 3 ⊢ ((𝐹‘𝑥)𝑅𝐶) ∈ V
2		eqid 2610	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))
3	1, 2	fnmpti 5935	. 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 2611	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
8	4, 5, 6, 7	ofcfval 29487	. . 3 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9	8	fneq1d 5895	. 2 ⊢ (𝜑 → ((𝐹∘𝑓/𝑐𝑅𝐶) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) Fn 𝐴))
10	3, 9	mpbiri 247	1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ∘_𝑓/𝑐cofc 29484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ofc 29485

This theorem is referenced by:  probfinmeasb  29818  coinflipspace  29869