Theorem ofcfval3 34208

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 3459	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3459	. . 3 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ V)
5		dmexg 7841	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		mptexg 7165	. . . 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 5852	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
11	9	fveq1d 6834	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑓‘𝑥) = (𝐹‘𝑥))
12		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
13	11, 12	oveq12d 7374	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
14	10, 13	mpteq12dv 5183	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15		df-ofc 34202	. . 3 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
16	14, 15	ovmpoga 7510	. 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 1541   ∈ wcel 2113  Vcvv 3438   ↦ cmpt 5177  dom cdm 5622  ‘cfv 6490  (class class class)co 7356   ∘_f/c cofc 34201

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-ofc 34202

This theorem is referenced by:  ofcfval4  34211  measdivcst  34330