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Theorem ssdmres 5379
Description: A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
Assertion
Ref Expression
ssdmres (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)

Proof of Theorem ssdmres
StepHypRef Expression
1 df-ss 3569 . 2 (𝐴 ⊆ dom 𝐵 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
2 dmres 5378 . . 3 dom (𝐵𝐴) = (𝐴 ∩ dom 𝐵)
32eqeq1i 2626 . 2 (dom (𝐵𝐴) = 𝐴 ↔ (𝐴 ∩ dom 𝐵) = 𝐴)
41, 3bitr4i 267 1 (𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cin 3554  wss 3555  dom cdm 5074  cres 5076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-dm 5084  df-res 5086
This theorem is referenced by:  dmresi  5416  fnssresb  5961  fores  6081  foimacnv  6111  dffv2  6228  sbthlem4  8017  hashimarn  13165  dvres3  23583  c1liplem1  23663  lhop1lem  23680  lhop  23683  trlreslem  26465  hhssabloi  27968  hhssnv  27970  hhshsslem1  27973  fresf1o  29277  exidreslem  33308  divrngcl  33388  isdrngo2  33389  dvbdfbdioolem1  39449  fourierdlem48  39678  fourierdlem49  39679  fourierdlem71  39701  fourierdlem73  39703  fourierdlem94  39724  fourierdlem111  39741  fourierdlem112  39742  fourierdlem113  39743  fouriersw  39755  fouriercn  39756  dmvon  40127
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