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Theorem fresin2 43739
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 6716 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
21eqcomd 2739 . . . 4 (𝐹:𝐴𝐵𝐴 = dom 𝐹)
32ineq2d 4210 . . 3 (𝐹:𝐴𝐵 → (𝐶𝐴) = (𝐶 ∩ dom 𝐹))
43reseq2d 5976 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹 ↾ (𝐶 ∩ dom 𝐹)))
5 frel 6712 . . 3 (𝐹:𝐴𝐵 → Rel 𝐹)
6 resindm 6025 . . 3 (Rel 𝐹 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
75, 6syl 17 . 2 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶 ∩ dom 𝐹)) = (𝐹𝐶))
84, 7eqtrd 2773 1 (𝐹:𝐴𝐵 → (𝐹 ↾ (𝐶𝐴)) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cin 3945  dom cdm 5672  cres 5674  Rel wrel 5677  wf 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-dm 5682  df-res 5684  df-fun 6537  df-fn 6538  df-f 6539
This theorem is referenced by: (None)
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