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Theorem fnresdmss 45629
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 6591 . 2 (𝐹 Fn 𝐴 → Rel 𝐹)
2 fndm 6592 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32adantr 482 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹 = 𝐴)
4 simpr 486 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐴𝐵)
53, 4eqsstrd 3951 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 relssres 5981 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
71, 5, 6syl2an2r 692 1 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1548  wss 3885  dom cdm 5621  cres 5623  Rel wrel 5626   Fn wfn 6484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-dm 5631  df-res 5633  df-fun 6491  df-fn 6492
This theorem is referenced by: (None)
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