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Theorem 1st2ndbr 5838
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 5835 . . 3 ((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 simpr 107 . . 3 ((Rel 𝐵𝐴𝐵) → 𝐴𝐵)
31, 2eqeltrrd 2131 . 2 ((Rel 𝐵𝐴𝐵) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝐵)
4 df-br 3793 . 2 ((1st𝐴)𝐵(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ 𝐵)
53, 4sylibr 141 1 ((Rel 𝐵𝐴𝐵) → (1st𝐴)𝐵(2nd𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  cop 3406   class class class wbr 3792  Rel wrel 4378  cfv 4930  1st c1st 5793  2nd c2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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