Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fresin2 Structured version   Visualization version   GIF version

Theorem fresin2 38347
 Description: Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
fresin2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))

Proof of Theorem fresin2
StepHypRef Expression
1 fdm 5964 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21eqcomd 2616 . . . 4 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
32ineq2d 3776 . . 3 (𝐹:𝐴𝐵 → (𝐶𝐴) = (𝐶 ∩ dom 𝐹))
43reseq2d 5317 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5 frel 5963 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
6 resindm 5364 . . 3 (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
75, 6syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
84, 7eqtrd 2644 1 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∩ cin 3539  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  ⟶wf 5800 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-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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator