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Theorem ofcval 31466
 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 2802 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
51, 2, 3, 4ofcfval 31465 . . . 4 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
65adantr 484 . . 3 ((𝜑𝑋𝐴) → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 simpr 488 . . . . 5 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
87fveq2d 6653 . . . 4 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 7154 . . 3 (((𝜑𝑋𝐴) ∧ 𝑥 = 𝑋) → ((𝐹𝑥)𝑅𝐶) = ((𝐹𝑋)𝑅𝐶))
10 simpr 488 . . 3 ((𝜑𝑋𝐴) → 𝑋𝐴)
11 ovexd 7174 . . 3 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) ∈ V)
126, 9, 10, 11fvmptd 6756 . 2 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = ((𝐹𝑋)𝑅𝐶))
13 ofcval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)
1413oveq1d 7154 . 2 ((𝜑𝑋𝐴) → ((𝐹𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
1512, 14eqtrd 2836 1 ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ↦ cmpt 5113   Fn wfn 6323  ‘cfv 6328  (class class class)co 7139   ∘f/c cofc 31462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-ofc 31463 This theorem is referenced by:  probfinmeasb  31794
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