Theorem ofcof 34300

Description: Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)

Hypotheses

Ref	Expression
ofcof.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofcof.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcof.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
ofcof	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶})))

Proof of Theorem ofcof

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcof.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6657	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcof.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcof.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
5		eqidd 2740	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
6	2, 3, 4, 5	ofcfval 34291	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		fnconstg 6716	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9		inidm 4156	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
10		fvconst2g 7147	. . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
11	4, 10	sylan 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
12	2, 8, 3, 3, 9, 5, 11	offval 7630	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
13	6, 12	eqtr4d 2777	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4556   ↦ cmpt 5154   × cxp 5617   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357   ∘_f cof 7619   ∘_f/c cofc 34288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-ofc 34289

This theorem is referenced by:  ofcccat  34736