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Theorem ofcfval4 34085
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 5242 . 2 (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
4 ofcfval4.2 . . . 4 (𝜑𝐴𝑉)
51, 4fexd 7246 . . 3 (𝜑𝐹 ∈ V)
6 ofcfval4.3 . . 3 (𝜑𝐶𝑊)
7 ofcfval3 34082 . . 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 7103 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
101feqmptd 6976 . . 3 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
11 eqidd 2735 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥𝐵 ↦ (𝑥𝑅𝐶)))
12 oveq1 7437 . . 3 (𝑥 = (𝐹𝑦) → (𝑥𝑅𝐶) = ((𝐹𝑦)𝑅𝐶))
139, 10, 11, 12fmptco 7148 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
143, 8, 133eqtr4d 2784 1 (𝜑 → (𝐹f/c 𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cmpt 5230  dom cdm 5688  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  f/c cofc 34075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-ofc 34076
This theorem is referenced by:  rrvmulc  34434
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