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Theorem ofcfval3 29942
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 3198 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 481 . 2 ((𝐹𝑉𝐶𝑊) → 𝐹 ∈ V)
3 elex 3198 . . 3 (𝐶𝑊𝐶 ∈ V)
43adantl 482 . 2 ((𝐹𝑉𝐶𝑊) → 𝐶 ∈ V)
5 dmexg 7044 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 mptexg 6438 . . . 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 473 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑓 = 𝐹)
109dmeqd 5286 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
119fveq1d 6150 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑓𝑥) = (𝐹𝑥))
12 simpr 477 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑐 = 𝐶)
1311, 12oveq12d 6622 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
1410, 13mpteq12dv 4693 . . 3 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
15 df-ofc 29936 . . 3 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
1614, 15ovmpt2ga 6743 . 2 ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
172, 4, 8, 16syl3anc 1323 1 ((𝐹𝑉𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cmpt 4673  dom cdm 5074  cfv 5847  (class class class)co 6604  𝑓/𝑐cofc 29935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-ofc 29936
This theorem is referenced by:  ofcfval4  29945  measdivcst  30066
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