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Theorem ofcof 33726
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 6723 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcof.2 . . 3 (𝜑𝐴𝑉)
4 ofcof.3 . . 3 (𝜑𝐶𝑊)
5 eqidd 2729 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
62, 3, 4, 5ofcfval 33717 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 fnconstg 6785 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
84, 7syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9 inidm 4219 . . 3 (𝐴𝐴) = 𝐴
10 fvconst2g 7214 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
114, 10sylan 579 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
122, 8, 3, 3, 9, 5, 11offval 7694 . 2 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
136, 12eqtr4d 2771 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {csn 4629  cmpt 5231   × cxp 5676   Fn wfn 6543  wf 6544  cfv 6548  (class class class)co 7420  f cof 7683  f/c cofc 33714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-ofc 33715
This theorem is referenced by:  ofcccat  34175
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