Theorem 1st2ndbr 5830

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) → (1st ‘𝐴)𝐵(2nd ‘𝐴))

Proof of Theorem 1st2ndbr

Step	Hyp	Ref	Expression
1		1st2nd 5827	. . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		simpr 108	. . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
3	1, 2	eqeltrrd 2156	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵)
4		df-br 3786	. 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵)
5	3, 4	sylibr 132	1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1433  ⟨cop 3401   class class class wbr 3785  Rel wrel 4368  ‘cfv 4922  1^st c1st 5785  2^nd c2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)