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Theorem fnresdmss 38342
 Description: A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
fnresdmss ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)

Proof of Theorem fnresdmss
StepHypRef Expression
1 fnrel 5903 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 fndm 5904 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43adantr 480 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹 = 𝐴)
5 simpr 476 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐴𝐵)
64, 5eqsstrd 3602 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
7 relssres 5357 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
82, 6, 7syl2anc 691 1 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  Rel wrel 5043   Fn wfn 5799 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 This theorem is referenced by: (None)
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