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Theorem ofcval 34080
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 2736 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
51, 2, 3, 4ofcfval 34079 . . . 4 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
65adantr 480 . . 3 ((𝜑𝑋𝐴) → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 simpr 484 . . . . 5 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
87fveq2d 6911 . . . 4 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 7446 . . 3 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑋)𝑅𝐶))
10 simpr 484 . . 3 ((𝜑𝑋𝐴) → 𝑋𝐴)
11 ovexd 7466 . . 3 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) ∈ V)
126, 9, 10, 11fvmptd 7023 . 2 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = ((𝐹𝑋)𝑅𝐶))
13 ofcval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)
1413oveq1d 7446 . 2 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
1512, 14eqtrd 2775 1 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cmpt 5231   Fn wfn 6558  cfv 6563  (class class class)co 7431  f/c cofc 34076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ofc 34077
This theorem is referenced by:  probfinmeasb  34410
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