Theorem ofcfval3 34613

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 3471	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 486	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3471	. . 3 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V)
4	3	adantl 487	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ V)
5		dmexg 7899	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		mptexg 7221	. . . 4 ⊢ (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
7	5, 6	syl 18	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
8	7	adantr 486	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
9		simpl 488	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑓 = 𝐹)
10	9	dmeqd 5889	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
11	9	fveq1d 6881	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑓‘𝑥) = (𝐹‘𝑥))
12		simpr 490	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
13	11, 12	oveq12d 7432	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
14	10, 13	mpteq12dv 5192	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15		df-ofc 34607	. . 3 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
16	14, 15	ovmpoga 7568	. 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 1398	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  dom cdm 5655  ‘cfv 6533  (class class class)co 7414   ∘_f/c cofc 34606

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-ofc 34607

This theorem is used by:  ofcfval4  34616  measdivcst  34736