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Theorem fresin2 39869
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 6212 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21eqcomd 2766 . . . 4 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
32ineq2d 3957 . . 3 (𝐹:𝐴𝐵 → (𝐶𝐴) = (𝐶 ∩ dom 𝐹))
43reseq2d 5551 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5 frel 6211 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
6 resindm 5602 . . 3 (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
75, 6syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
84, 7eqtrd 2794 1 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cin 3714  dom cdm 5266  cres 5268  Rel wrel 5271  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by: (None)
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