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Theorem ofcval 29488
 Description: Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1 (𝜑𝐹 Fn 𝐴)
ofcfval.2 (𝜑𝐴𝑉)
ofcfval.3 (𝜑𝐶𝑊)
ofcval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)
Assertion
Ref Expression
ofcval ((𝜑𝑋𝐴) → ((𝐹𝑓/𝑐𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Proof of Theorem ofcval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcfval.1 . . . . 5 (𝜑𝐹 Fn 𝐴)
2 ofcfval.2 . . . . 5 (𝜑𝐴𝑉)
3 ofcfval.3 . . . . 5 (𝜑𝐶𝑊)
4 eqidd 2611 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
51, 2, 3, 4ofcfval 29487 . . . 4 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
65adantr 480 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 simpr 476 . . . . 5 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
87fveq2d 6107 . . . 4 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 6564 . . 3 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑋)𝑅𝐶))
10 simpr 476 . . 3 ((𝜑𝑋𝐴) → 𝑋𝐴)
11 ovex 6577 . . . 4 ((𝐹𝑋)𝑅𝐶) ∈ V
1211a1i 11 . . 3 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) ∈ V)
136, 9, 10, 12fvmptd 6197 . 2 ((𝜑𝑋𝐴) → ((𝐹𝑓/𝑐𝑅𝐶)‘𝑋) = ((𝐹𝑋)𝑅𝐶))
14 ofcval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)
1514oveq1d 6564 . 2 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
1613, 15eqtrd 2644 1 ((𝜑𝑋𝐴) → ((𝐹𝑓/𝑐𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ∘𝑓/𝑐cofc 29484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ofc 29485 This theorem is referenced by:  probfinmeasb  29818
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