Theorem ofcfval3 30512

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

Assertion

Ref	Expression
ofcfval3	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹∘𝑓/𝑐𝑅𝐶) = (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 3417	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 468	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3417	. . 3 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V)
4	3	adantl 469	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ V)
5		dmexg 7337	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		mptexg 6719	. . . 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 468	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V)
9		simpl 470	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑓 = 𝐹)
10	9	dmeqd 5541	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
11	9	fveq1d 6420	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑓‘𝑥) = (𝐹‘𝑥))
12		simpr 473	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
13	11, 12	oveq12d 6902	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶))
14	10, 13	mpteq12dv 4938	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15		df-ofc 30506	. . 3 ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
16	14, 15	ovmpt2ga 7030	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))
17	2, 4, 8, 16	syl3anc 1483	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹∘𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2157  Vcvv 3402   ↦ cmpt 4934  dom cdm 5324  ‘cfv 6111  (class class class)co 6884  ∘_𝑓/𝑐cofc 30505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-un 7189

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-ofc 30506

This theorem is referenced by:  ofcfval4  30515  measdivcst  30636