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Theorem ofcfval3 30512
Description: General value of (𝐹𝑓/𝑐𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
ofcfval3 ((𝐹𝑉𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (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 3417 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 468 . 2 ((𝐹𝑉𝐶𝑊) → 𝐹 ∈ V)
3 elex 3417 . . 3 (𝐶𝑊𝐶 ∈ V)
43adantl 469 . 2 ((𝐹𝑉𝐶𝑊) → 𝐶 ∈ V)
5 dmexg 7337 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 mptexg 6719 . . . 4 (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
75, 6syl 17 . . 3 (𝐹𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
87adantr 468 . 2 ((𝐹𝑉𝐶𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
9 simpl 470 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑓 = 𝐹)
109dmeqd 5541 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
119fveq1d 6420 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑓𝑥) = (𝐹𝑥))
12 simpr 473 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑐 = 𝐶)
1311, 12oveq12d 6902 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
1410, 13mpteq12dv 4938 . . 3 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
15 df-ofc 30506 . . 3 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
1614, 15ovmpt2ga 7030 . 2 ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
172, 4, 8, 16syl3anc 1483 1 ((𝐹𝑉𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
Colors of variables: wff setvar class
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
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