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

Theorem brresg 5558
Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
Assertion
Ref Expression
brresg (𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))

Proof of Theorem brresg
StepHypRef Expression
1 opelresg 5557 . 2 (𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
2 df-br 4805 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
3 df-br 4805 . . 3 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
43anbi1i 733 . 2 ((𝐴𝐶𝐵𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
51, 2, 43bitr4g 303 1 (𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  cop 4327   class class class wbr 4804  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278
This theorem is referenced by:  brres  5560  fv1stcnv  31985  fv2ndcnv  31986
  Copyright terms: Public domain W3C validator