MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relssres Structured version   Visualization version   GIF version

Theorem relssres 6013
Description: Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
relssres ((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)

Proof of Theorem relssres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . 4 ((Rel 𝐴 ∧ dom 𝐴𝐵) → Rel 𝐴)
2 vex 3470 . . . . . . . . 9 𝑥 ∈ V
3 vex 3470 . . . . . . . . 9 𝑦 ∈ V
42, 3opeldm 5898 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
5 ssel 3968 . . . . . . . 8 (dom 𝐴𝐵 → (𝑥 ∈ dom 𝐴𝑥𝐵))
64, 5syl5 34 . . . . . . 7 (dom 𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥𝐵))
76ancrd 551 . . . . . 6 (dom 𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
83opelresi 5980 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ (𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
97, 8imbitrrdi 251 . . . . 5 (dom 𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵)))
109adantl 481 . . . 4 ((Rel 𝐴 ∧ dom 𝐴𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵)))
111, 10relssdv 5779 . . 3 ((Rel 𝐴 ∧ dom 𝐴𝐵) → 𝐴 ⊆ (𝐴𝐵))
12 resss 5997 . . 3 (𝐴𝐵) ⊆ 𝐴
1311, 12jctil 519 . 2 ((Rel 𝐴 ∧ dom 𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐴𝐴 ⊆ (𝐴𝐵)))
14 eqss 3990 . 2 ((𝐴𝐵) = 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴𝐴 ⊆ (𝐴𝐵)))
1513, 14sylibr 233 1 ((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wss 3941  cop 4627  dom cdm 5667  cres 5669  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-dm 5677  df-res 5679
This theorem is referenced by:  resdm  6017  fnresdm  6660  focofo  6809  f1ompt  7103  tfr2b  8392  tz7.48-2  8438  omxpenlem  9070  pwfir  9173  rankwflemb  9785  zorn2lem4  10491  relexpaddg  14998  setscom  17114  setsid  17142  dprd2da  19956  dprd2db  19957  ustssco  24043  dvres3  25766  dvres3a  25767  rlimcnp2  26817  nolt02o  27547  nogt01o  27548  nosupbnd1  27566  noinfbnd1  27581  ex-res  30166  symgcom2  32716  poimirlem3  36985  relexpaddss  42983  fnresdmss  44377  limsupresuz  44929  liminfresuz  45010
  Copyright terms: Public domain W3C validator