Theorem 1st2ndbr 6184

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 6181	. . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		simpr 110	. . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
3	1, 2	eqeltrrd 2255	. 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵)
4		df-br 4004	. 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵)
5	3, 4	sylibr 134	1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148  ⟨cop 3595   class class class wbr 4003  Rel wrel 4631  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fv 5224  df-1st 6140  df-2nd 6141

This theorem is referenced by: (None)