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Theorem ofcval 32365
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/c 𝑅𝐶)‘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 2737 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
51, 2, 3, 4ofcfval 32364 . . . 4 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
65adantr 481 . . 3 ((𝜑𝑋𝐴) → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 simpr 485 . . . . 5 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
87fveq2d 6829 . . . 4 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 7352 . . 3 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑋)𝑅𝐶))
10 simpr 485 . . 3 ((𝜑𝑋𝐴) → 𝑋𝐴)
11 ovexd 7372 . . 3 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) ∈ V)
126, 9, 10, 11fvmptd 6938 . 2 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = ((𝐹𝑋)𝑅𝐶))
13 ofcval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)
1413oveq1d 7352 . 2 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
1512, 14eqtrd 2776 1 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  cmpt 5175   Fn wfn 6474  cfv 6479  (class class class)co 7337  f/c cofc 32361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-ofc 32362
This theorem is referenced by:  probfinmeasb  32695
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