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Theorem ofcfval3 31260
Description: General value of (𝐹f/c 𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
ofcfval3 ((𝐹𝑉𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)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.
StepHypRef Expression
1 elex 3510 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 481 . 2 ((𝐹𝑉𝐶𝑊) → 𝐹 ∈ V)
3 elex 3510 . . 3 (𝐶𝑊𝐶 ∈ V)
43adantl 482 . 2 ((𝐹𝑉𝐶𝑊) → 𝐶 ∈ V)
5 dmexg 7602 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 mptexg 6975 . . . 4 (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
75, 6syl 17 . . 3 (𝐹𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
87adantr 481 . 2 ((𝐹𝑉𝐶𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
9 simpl 483 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑓 = 𝐹)
109dmeqd 5767 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
119fveq1d 6665 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑓𝑥) = (𝐹𝑥))
12 simpr 485 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑐 = 𝐶)
1311, 12oveq12d 7163 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
1410, 13mpteq12dv 5142 . . 3 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
15 df-ofc 31254 . . 3 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
1614, 15ovmpoga 7293 . 2 ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
172, 4, 8, 16syl3anc 1363 1 ((𝐹𝑉𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  dom cdm 5548  cfv 6348  (class class class)co 7145  f/c cofc 31253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ofc 31254
This theorem is referenced by:  ofcfval4  31263  measdivcst  31382
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