Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcof Structured version   Visualization version   GIF version

Theorem ofcof 30297
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 6083 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
4 ofcof.2 . . 3 (𝜑𝐴𝑉)
5 ofcof.3 . . 3 (𝜑𝐶𝑊)
6 eqidd 2652 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
73, 4, 5, 6ofcfval 30288 . 2 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
8 fnconstg 6131 . . . 4 (𝐶𝑊 → (𝐴 × {𝐶}) Fn 𝐴)
95, 8syl 17 . . 3 (𝜑 → (𝐴 × {𝐶}) Fn 𝐴)
10 inidm 3855 . . 3 (𝐴𝐴) = 𝐴
11 fvconst2g 6508 . . . 4 ((𝐶𝑊𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
125, 11sylan 487 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶)
133, 9, 4, 4, 10, 6, 12offval 6946 . 2 (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅𝐶)))
147, 13eqtr4d 2688 1 (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝐹𝑓 𝑅(𝐴 × {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {csn 4210  cmpt 4762   × cxp 5141   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑓/𝑐cofc 30285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofc 30286
This theorem is referenced by:  ofcccat  30748
  Copyright terms: Public domain W3C validator