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Theorem ofcfval3 30498
 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 ((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 3361 . . 3 (𝐹𝑉𝐹 ∈ V)
21adantr 466 . 2 ((𝐹𝑉𝐶𝑊) → 𝐹 ∈ V)
3 elex 3361 . . 3 (𝐶𝑊𝐶 ∈ V)
43adantl 467 . 2 ((𝐹𝑉𝐶𝑊) → 𝐶 ∈ V)
5 dmexg 7243 . . . 4 (𝐹𝑉 → dom 𝐹 ∈ V)
6 mptexg 6627 . . . 4 (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
75, 6syl 17 . . 3 (𝐹𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
87adantr 466 . 2 ((𝐹𝑉𝐶𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
9 simpl 468 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑓 = 𝐹)
109dmeqd 5464 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → dom 𝑓 = dom 𝐹)
119fveq1d 6334 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑓𝑥) = (𝐹𝑥))
12 simpr 471 . . . . 5 ((𝑓 = 𝐹𝑐 = 𝐶) → 𝑐 = 𝐶)
1311, 12oveq12d 6810 . . . 4 ((𝑓 = 𝐹𝑐 = 𝐶) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
1410, 13mpteq12dv 4865 . . 3 ((𝑓 = 𝐹𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
15 df-ofc 30492 . . 3 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
1614, 15ovmpt2ga 6936 . 2 ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
172, 4, 8, 16syl3anc 1475 1 ((𝐹𝑉𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ↦ cmpt 4861  dom cdm 5249  ‘cfv 6031  (class class class)co 6792  ∘𝑓/𝑐cofc 30491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-ofc 30492 This theorem is referenced by:  ofcfval4  30501  measdivcst  30622
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