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Theorem resdmdfsn 5365
 Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)
Assertion
Ref Expression
resdmdfsn (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

Proof of Theorem resdmdfsn
StepHypRef Expression
1 indif1 3830 . . . 4 ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
2 incom 3767 . . . . . 6 (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
3 inv1 3922 . . . . . 6 (dom 𝑅 ∩ V) = dom 𝑅
42, 3eqtri 2632 . . . . 5 (V ∩ dom 𝑅) = dom 𝑅
54difeq1i 3686 . . . 4 ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
61, 5eqtri 2632 . . 3 ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
76reseq2i 5314 . 2 (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
8 resindm 5364 . 2 (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
97, 8syl5reqr 2659 1 (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539  {csn 4125  dom cdm 5038   ↾ cres 5040  Rel wrel 5043 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 This theorem is referenced by:  funresdfunsn  6360
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