Theorem ofcfval3 34069

Description: General value of (𝐹 ∘_f/c 𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Assertion

Ref	Expression
ofcfval3	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))

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

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

Proof of Theorem ofcfval3

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

Step	Hyp	Ref	Expression
1		elex 3457	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3457	. . 3 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ V)
5		dmexg 7834	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		mptexg 7157	. . . 4 ⊢ (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
8	7	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
9		simpl 482	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑓 = 𝐹)
10	9	dmeqd 5848	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
11	9	fveq1d 6824	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑓‘𝑥) = (𝐹‘𝑥))
12		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
13	11, 12	oveq12d 7367	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
14	10, 13	mpteq12dv 5179	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15		df-ofc 34063	. . 3 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
16	14, 15	ovmpoga 7503	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
17	2, 4, 8, 16	syl3anc 1373	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ↦ cmpt 5173  dom cdm 5619  ‘cfv 6482  (class class class)co 7349   ∘_f/c cofc 34062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  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 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-ofc 34063

This theorem is referenced by:  ofcfval4  34072  measdivcst  34191