Theorem ofcfval 34106

Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofcfval.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfval.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)
ofcfval.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Assertion

Ref	Expression
ofcfval	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofcfval

Dummy variables 𝑓 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofc 34104	. . . 4 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
2	1	a1i 11	. . 3 ⊢ (𝜑 → ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))))
3		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹)
4	3	dmeqd 5845	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
5	3	fveq1d 6824	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥))
6		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶)
7	5, 6	oveq12d 7364	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
8	4, 7	mpteq12dv 5178	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
9		ofcfval.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
10		ofcfval.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
11		fnex 7151	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
13		ofcfval.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
14	13	elexd 3460	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
15	9	fndmd 6586	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15, 10	eqeltrd 2831	. . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉)
17	16	mptexd 7158	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
18	2, 8, 12, 14, 17	ovmpod 7498	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
19	15	eleq2d 2817	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
20	19	pm5.32i 574	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))
21		ofcfval.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
22	20, 21	sylbi 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵)
23	22	oveq1d 7361	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
24	15, 23	mpteq12dva 5177	. 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
25	18, 24	eqtrd 2766	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ↦ cmpt 5172  dom cdm 5616   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   ∘_f/c cofc 34103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-ofc 34104

This theorem is referenced by:  ofcval  34107  ofcfn  34108  ofcfeqd2  34109  ofcf  34111  ofcfval2  34112  ofcc  34114  ofcof  34115