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Theorem ofcfval 34099
Description: Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1 (𝜑𝐹 Fn 𝐴)
ofcfval.2 (𝜑𝐴𝑉)
ofcfval.3 (𝜑𝐶𝑊)
ofcfval.6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Assertion
Ref Expression
ofcfval (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofcfval
Dummy variables 𝑓 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofc 34097 . . . 4 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
21a1i 11 . . 3 (𝜑 → ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))))
3 simprl 771 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑓 = 𝐹)
43dmeqd 5916 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → dom 𝑓 = dom 𝐹)
53fveq1d 6908 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑓𝑥) = (𝐹𝑥))
6 simprr 773 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → 𝑐 = 𝐶)
75, 6oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → ((𝑓𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅𝐶))
84, 7mpteq12dv 5233 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑐 = 𝐶)) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
9 ofcfval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
10 ofcfval.2 . . . 4 (𝜑𝐴𝑉)
11 fnex 7237 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
129, 10, 11syl2anc 584 . . 3 (𝜑𝐹 ∈ V)
13 ofcfval.3 . . . 4 (𝜑𝐶𝑊)
1413elexd 3504 . . 3 (𝜑𝐶 ∈ V)
159fndmd 6673 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
1615, 10eqeltrd 2841 . . . 4 (𝜑 → dom 𝐹𝑉)
1716mptexd 7244 . . 3 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) ∈ V)
182, 8, 12, 14, 17ovmpod 7585 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
1915eleq2d 2827 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2019pm5.32i 574 . . . . 5 ((𝜑𝑥 ∈ dom 𝐹) ↔ (𝜑𝑥𝐴))
21 ofcfval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2220, 21sylbi 217 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = 𝐵)
2322oveq1d 7446 . . 3 ((𝜑𝑥 ∈ dom 𝐹) → ((𝐹𝑥)𝑅𝐶) = (𝐵𝑅𝐶))
2415, 23mpteq12dva 5231 . 2 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
2518, 24eqtrd 2777 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  dom cdm 5685   Fn wfn 6556  cfv 6561  (class class class)co 7431  cmpo 7433  f/c cofc 34096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ofc 34097
This theorem is referenced by:  ofcval  34100  ofcfn  34101  ofcfeqd2  34102  ofcf  34104  ofcfval2  34105  ofcc  34107  ofcof  34108
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