Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  1st2ndbr Structured version   GIF version

Theorem 1st2ndbr 5729
 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 B A B) → (1stA)B(2ndA))

Proof of Theorem 1st2ndbr
StepHypRef Expression
1 1st2nd 5726 . . 3 ((Rel B A B) → A = ⟨(1stA), (2ndA)⟩)
2 simpr 103 . . 3 ((Rel B A B) → A B)
31, 2eqeltrrd 2093 . 2 ((Rel B A B) → ⟨(1stA), (2ndA)⟩ B)
4 df-br 3735 . 2 ((1stA)B(2ndA) ↔ ⟨(1stA), (2ndA)⟩ B)
53, 4sylibr 137 1 ((Rel B A B) → (1stA)B(2ndA))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1370  ⟨cop 3349   class class class wbr 3734  Rel wrel 4273  ‘cfv 4825  1st c1st 5684  2nd c2nd 5685 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fv 4833  df-1st 5686  df-2nd 5687 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator