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Theorem ofcfval4 30507
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/𝑐𝑅𝐶) = ((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 6193 . . 3 (𝜑 → dom 𝐹 = 𝐴)
32mpteq1d 4873 . 2 (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
4 ofcfval4.2 . . . 4 (𝜑𝐴𝑉)
5 fex 6636 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
61, 4, 5syl2anc 573 . . 3 (𝜑𝐹 ∈ V)
7 ofcfval4.3 . . 3 (𝜑𝐶𝑊)
8 ofcfval3 30504 . . 3 ((𝐹 ∈ V ∧ 𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
96, 7, 8syl2anc 573 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹𝑦)𝑅𝐶)))
101ffvelrnda 6504 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
111feqmptd 6393 . . 3 (𝜑𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
12 eqidd 2772 . . 3 (𝜑 → (𝑥𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥𝐵 ↦ (𝑥𝑅𝐶)))
13 oveq1 6803 . . 3 (𝑥 = (𝐹𝑦) → (𝑥𝑅𝐶) = ((𝐹𝑦)𝑅𝐶))
1410, 11, 12, 13fmptco 6542 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦𝐴 ↦ ((𝐹𝑦)𝑅𝐶)))
153, 9, 143eqtr4d 2815 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4864  dom cdm 5250  ccom 5254  wf 6026  cfv 6030  (class class class)co 6796  𝑓/𝑐cofc 30497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-ofc 30498
This theorem is referenced by:  rrvmulc  30855
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