Theorem ofcfval 32062

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 32060	. . . 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 768	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑓 = 𝐹)
4	3	dmeqd 5813	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
5	3	fveq1d 6773	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → (𝑓‘𝑥) = (𝐹‘𝑥))
6		simprr 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → 𝑐 = 𝐶)
7	5, 6	oveq12d 7289	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
8	4, 7	mpteq12dv 5170	. . 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 7090	. . . 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 3451	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
15	9	fndmd 6536	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15, 10	eqeltrd 2841	. . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉)
17	16	mptexd 7097	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
18	2, 8, 12, 14, 17	ovmpod 7419	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
19	15	eleq2d 2826	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
20	19	pm5.32i 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))
21		ofcfval.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
22	20, 21	sylbi 216	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 𝐵)
23	22	oveq1d 7286	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
24	15, 23	mpteq12dva 5168	. 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
25	18, 24	eqtrd 2780	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ↦ cmpt 5162  dom cdm 5590   Fn wfn 6427  ‘cfv 6432  (class class class)co 7271   ∈ cmpo 7273   ∘_f/c cofc 32059

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-ofc 32060

This theorem is referenced by:  ofcval  32063  ofcfn  32064  ofcfeqd2  32065  ofcf  32067  ofcfval2  32068  ofcc  32070  ofcof  32071