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Theorem 1st2ndbr 6126
 Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
1st2ndbr ((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))

Proof of Theorem 1st2ndbr
StepHypRef Expression
1 1st2nd 6123 . . 3 ((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 simpr 109 . . 3 ((Rel 𝐵𝐴𝐵) → 𝐴𝐵)
31, 2eqeltrrd 2235 . 2 ((Rel 𝐵𝐴𝐵) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝐵)
4 df-br 3966 . 2 ((1st𝐴)𝐵(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝐵)
53, 4sylibr 133 1 ((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  Rel wrel 4588  ‘cfv 5167  1st c1st 6080  2nd c2nd 6081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-1st 6082  df-2nd 6083 This theorem is referenced by: (None)
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