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Theorem ofcof 31474
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 6492 . . 3 (𝜑𝐹 Fn 𝐴)
3 ofcof.2 . . 3 (𝜑𝐴𝑉)
4 ofcof.3 . . 3 (𝜑𝐶𝑊)
5 eqidd 2802 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
62, 3, 4, 5ofcfval 31465 . 2 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
7 fnconstg 6545 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
84, 7syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
9 inidm 4148 . . 3 (𝐴𝐴) = 𝐴
10 fvconst2g 6945 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
114, 10sylan 583 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
122, 8, 3, 3, 9, 5, 11offval 7400 . 2 (𝜑 → (𝐹f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
136, 12eqtr4d 2839 1 (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {csn 4528  cmpt 5113   × cxp 5521   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391  f/c cofc 31462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofc 31463
This theorem is referenced by:  ofcccat  31921
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