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Theorem fnresdmss 39164
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 5977 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 481 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 fndm 5978 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43adantr 481 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹 = 𝐴)
5 simpr 477 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐴𝐵)
64, 5eqsstrd 3631 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
7 relssres 5425 . 2 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
82, 6, 7syl2anc 692 1 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐵) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wss 3567  dom cdm 5104  cres 5106  Rel wrel 5109   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-dm 5114  df-res 5116  df-fun 5878  df-fn 5879
This theorem is referenced by: (None)
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