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Theorem fnresdmss 45746
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 6623 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 6624 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32adantr 484 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹 = 𝐴)
4 simpr 488 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐴𝐵)
53, 4eqsstrd 3970 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 relssres 6008 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
71, 5, 6syl2an2r 695 1 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1560  wss 3904  dom cdm 5647  cres 5649  Rel wrel 5652   Fn wfn 6516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-dm 5657  df-res 5659  df-fun 6523  df-fn 6524
This theorem is referenced by: (None)
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