Theorem ofcfval 34132

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 34130	. . . 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 5849	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
5	3	fveq1d 6830	. . . . 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 7370	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑐 = 𝐶)) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
8	4, 7	mpteq12dv 5180	. . 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 7157	. . . 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 3461	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
15	9	fndmd 6591	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15, 10	eqeltrd 2833	. . . 4 ⊢ (𝜑 → dom 𝐹 ∈ 𝑉)
17	16	mptexd 7164	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
18	2, 8, 12, 14, 17	ovmpod 7504	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
19	15	eleq2d 2819	. . . . . 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 7367	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
24	15, 23	mpteq12dva 5179	. 2 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
25	18, 24	eqtrd 2768	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ↦ cmpt 5174  dom cdm 5619   Fn wfn 6481  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354   ∘_f/c cofc 34129

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  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 7355  df-oprab 7356  df-mpo 7357  df-ofc 34130

This theorem is referenced by:  ofcval  34133  ofcfn  34134  ofcfeqd2  34135  ofcf  34137  ofcfval2  34138  ofcc  34140  ofcof  34141