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Theorem ofcfval4 32069
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 6609 . . 3 (𝜑 → dom 𝐹 = 𝐴)
32mpteq1d 5174 . 2 (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
4 ofcfval4.2 . . . 4 (𝜑𝐴𝑉)
51, 4fexd 7100 . . 3 (𝜑𝐹 ∈ V)
6 ofcfval4.3 . . 3 (𝜑𝐶𝑊)
7 ofcfval3 32066 . . 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)))
91ffvelrnda 6958 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
101feqmptd 6834 . . 3 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
11 eqidd 2741 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥𝐵 ↦ (𝑥𝑅𝐶)))
12 oveq1 7278 . . 3 (𝑥 = (𝐹𝑦) → (𝑥𝑅𝐶) = ((𝐹𝑦)𝑅𝐶))
139, 10, 11, 12fmptco 6998 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
143, 8, 133eqtr4d 2790 1 (𝜑 → (𝐹f/c 𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  Vcvv 3431  cmpt 5162  dom cdm 5590  ccom 5594  wf 6428  cfv 6432  (class class class)co 7271  f/c cofc 32059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-ofc 32060
This theorem is referenced by:  rrvmulc  32416
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