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Theorem ofcfval4 32744
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 6684 . . 3 (𝜑 → dom 𝐹 = 𝐴)
32mpteq1d 5205 . 2 (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
4 ofcfval4.2 . . . 4 (𝜑𝐴𝑉)
51, 4fexd 7182 . . 3 (𝜑𝐹 ∈ V)
6 ofcfval4.3 . . 3 (𝜑𝐶𝑊)
7 ofcfval3 32741 . . 3 ((𝐹 ∈ V ∧ 𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
85, 6, 7syl2anc 585 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
91ffvelcdmda 7040 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
101feqmptd 6915 . . 3 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
11 eqidd 2738 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥𝐵 ↦ (𝑥𝑅𝐶)))
12 oveq1 7369 . . 3 (𝑥 = (𝐹𝑦) → (𝑥𝑅𝐶) = ((𝐹𝑦)𝑅𝐶))
139, 10, 11, 12fmptco 7080 . 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 1542  wcel 2107  Vcvv 3448  cmpt 5193  dom cdm 5638  ccom 5642  wf 6497  cfv 6501  (class class class)co 7362  f/c cofc 32734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-ofc 32735
This theorem is referenced by:  rrvmulc  33093
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