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Theorem ofcfval3 31475
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 3462 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 484 . 2 ((𝐹𝑉𝐶𝑊) → 𝐹 ∈ V)
3 elex 3462 . . 3 (𝐶𝑊𝐶 ∈ V)
43adantl 485 . 2 ((𝐹𝑉𝐶𝑊) → 𝐶 ∈ V)
5 dmexg 7598 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 mptexg 6965 . . . 4 (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
75, 6syl 17 . . 3 (𝐹𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
87adantr 484 . 2 ((𝐹𝑉𝐶𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
9 simpl 486 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑓 = 𝐹)
109dmeqd 5742 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
119fveq1d 6651 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑓𝑥) = (𝐹𝑥))
12 simpr 488 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑐 = 𝐶)
1311, 12oveq12d 7157 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
1410, 13mpteq12dv 5118 . . 3 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
15 df-ofc 31469 . . 3 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
1614, 15ovmpoga 7287 . 2 ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
172, 4, 8, 16syl3anc 1368 1 ((𝐹𝑉𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cmpt 5113  dom cdm 5523  cfv 6328  (class class class)co 7139  f/c cofc 31468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-ofc 31469
This theorem is referenced by:  ofcfval4  31478  measdivcst  31597
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