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Theorem fresin2 41723
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 6511 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21eqcomd 2830 . . . 4 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
32ineq2d 4174 . . 3 (𝐹:𝐴𝐵 → (𝐶𝐴) = (𝐶 ∩ dom 𝐹))
43reseq2d 5840 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5 frel 6508 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
6 resindm 5887 . . 3 (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
75, 6syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
84, 7eqtrd 2859 1 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cin 3918  dom cdm 5542  cres 5544  Rel wrel 5547  wf 6339
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-dm 5552  df-res 5554  df-fun 6345  df-fn 6346  df-f 6347
This theorem is referenced by: (None)
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