Theorem ofcof 33093

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 6715	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcof.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcof.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
5		eqidd 2733	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
6	2, 3, 4, 5	ofcfval 33084	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		fnconstg 6776	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9		inidm 4217	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
10		fvconst2g 7199	. . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
11	4, 10	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
12	2, 8, 3, 3, 9, 5, 11	offval 7675	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
13	6, 12	eqtr4d 2775	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {csn 4627   ↦ cmpt 5230   × cxp 5673   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405   ∘_f cof 7664   ∘_f/c cofc 33081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofc 33082

This theorem is referenced by:  ofcccat  33542