Theorem ofcof 31016

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/𝑐𝑅𝐶) = (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 6345	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		ofcof.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofcof.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
5		eqidd 2779	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
6	2, 3, 4, 5	ofcfval 31007	. 2 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		fnconstg 6396	. . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9		inidm 4082	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
10		fvconst2g 6791	. . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
11	4, 10	sylan 572	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
12	2, 8, 3, 3, 9, 5, 11	offval 7234	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
13	6, 12	eqtr4d 2817	1 ⊢ (𝜑 → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝐹 ∘𝑓 𝑅(𝐴 × {𝐶})))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {csn 4441   ↦ cmpt 5008   × cxp 5405   Fn wfn 6183  ⟶wf 6184  ‘cfv 6188  (class class class)co 6976   ∘_𝑓 cof 7225  ∘_𝑓/𝑐cofc 31004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-of 7227  df-ofc 31005

This theorem is referenced by:  ofcccat  31465