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Theorem ofcfval4 34106
Description: The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval4.1 (𝜑𝐹:𝐴𝐵)
ofcfval4.2 (𝜑𝐴𝑉)
ofcfval4.3 (𝜑𝐶𝑊)
Assertion
Ref Expression
ofcfval4 (𝜑 → (𝐹f/c 𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofcfval4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ofcfval4.1 . . . 4 (𝜑𝐹:𝐴𝐵)
21fdmd 6746 . . 3 (𝜑 → dom 𝐹 = 𝐴)
32mpteq1d 5237 . 2 (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
4 ofcfval4.2 . . . 4 (𝜑𝐴𝑉)
51, 4fexd 7247 . . 3 (𝜑𝐹 ∈ V)
6 ofcfval4.3 . . 3 (𝜑𝐶𝑊)
7 ofcfval3 34103 . . 3 ((𝐹 ∈ V ∧ 𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
85, 6, 7syl2anc 584 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
91ffvelcdmda 7104 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
101feqmptd 6977 . . 3 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
11 eqidd 2738 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥𝐵 ↦ (𝑥𝑅𝐶)))
12 oveq1 7438 . . 3 (𝑥 = (𝐹𝑦) → (𝑥𝑅𝐶) = ((𝐹𝑦)𝑅𝐶))
139, 10, 11, 12fmptco 7149 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
143, 8, 133eqtr4d 2787 1 (𝜑 → (𝐹f/c 𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225  dom cdm 5685  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  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  ax-un 7755
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:  rrvmulc  34455
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