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Theorem ofcof 34088
Description: Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)
Hypotheses
Ref Expression
ofcof.1 (𝜑𝐹:𝐴𝐵)
ofcof.2 (𝜑𝐴𝑉)
ofcof.3 (𝜑𝐶𝑊)
Assertion
Ref Expression
ofcof (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))

Proof of Theorem ofcof
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcof.1 . . . 4 (𝜑𝐹:𝐴𝐵)
21ffnd 6738 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcof.2 . . 3 (𝜑𝐴𝑉)
4 ofcof.3 . . 3 (𝜑𝐶𝑊)
5 eqidd 2736 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
62, 3, 4, 5ofcfval 34079 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 fnconstg 6797 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
84, 7syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9 inidm 4235 . . 3 (𝐴𝐴) = 𝐴
10 fvconst2g 7222 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
114, 10sylan 580 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
122, 8, 3, 3, 9, 5, 11offval 7706 . 2 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
136, 12eqtr4d 2778 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {csn 4631  cmpt 5231   × cxp 5687   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  f/c cofc 34076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofc 34077
This theorem is referenced by:  ofcccat  34537
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