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Theorem ofcof 29496
 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/𝑐𝑅𝐶) = (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)
2 ffn 5958 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
4 ofcof.2 . . 3 (𝜑𝐴𝑉)
5 ofcof.3 . . 3 (𝜑𝐶𝑊)
6 eqidd 2611 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
73, 4, 5, 6ofcfval 29487 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
8 fnconstg 6006 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
95, 8syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
11 fvconst2g 6372 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
125, 11sylan 487 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
133, 9, 4, 4, 10, 6, 12offval 6802 . 2 (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
147, 13eqtr4d 2647 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝐹𝑓 𝑅(𝐴 × {𝐶})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125   ↦ cmpt 4643   × cxp 5036   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ∘𝑓/𝑐cofc 29484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofc 29485 This theorem is referenced by:  ofcccat  29946
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