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Theorem ofcof 33624
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 6709 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcof.2 . . 3 (𝜑𝐴𝑉)
4 ofcof.3 . . 3 (𝜑𝐶𝑊)
5 eqidd 2725 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
62, 3, 4, 5ofcfval 33615 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 fnconstg 6770 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
84, 7syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9 inidm 4211 . . 3 (𝐴𝐴) = 𝐴
10 fvconst2g 7196 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
114, 10sylan 579 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
122, 8, 3, 3, 9, 5, 11offval 7673 . 2 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
136, 12eqtr4d 2767 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {csn 4621  cmpt 5222   × cxp 5665   Fn wfn 6529  wf 6530  cfv 6534  (class class class)co 7402  f cof 7662  f/c cofc 33612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-ofc 33613
This theorem is referenced by:  ofcccat  34073
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